Luis Ribes Profi Nite Graphs and Groups Ergebnisse Der Mathematik Und Volume 66 Ihrer Grenzgebiete

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge A Series of Modern Surveys in Mathematics 66

Luis Ribes Profi nite Graphs and Groups Ergebnisse der Mathematik und Volume 66 ihrer Grenzgebiete

3. Folge

A Series of Modern Surveys in Mathematics

Editorial Board L. Ambrosio, Pisa V. Baladi, Paris G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette G. Huisken, Tübingen J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay U. Tillmann, Oxford J. Tits, Paris D.B. Zagier, Bonn

For further volumes: www.springer.com/series/728 Luis Ribes

Proﬁnite Graphs and Groups Luis Ribes School of Mathematics and Statistics Carleton University Ottawa, ON, Canada

ISSN 0071-1136 ISSN 2197-5655 (electronic) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-319-61041-2 ISBN 978-3-319-61199-0 (eBook) DOI 10.1007/978-3-319-61199-0

Library of Congress Control Number: 2017952053

Mathematics Subject Classiﬁcation: 20E18, 20E06, 20E08, 20F65, 20J05, 05C05, 20M35, 22C05

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Para Luisito y Tomasín Preface

Profinite groups are Galois groups, which we view as topological groups. In this book the theory of profinite graphs is developed as a natural tool in the study of some aspects of profinite and abstract groups. Our approach is modelled on the by now classical BassÐSerre theory of abstract groups acting on abstract trees as it appears in J.-P. Serre’s monograph ‘Trees’. We think of a graph Γ as the union of its sets of vertices V and edges E. A graph Γ is profinite if it is endowed with a profinite topology (i.e., a compact, Hausdorff and totally disconnected topology), in such a way that the functions defining the origin and terminal points are continuous. A natural example of a profinite graph is the Cayley graph Γ(G,X)of a profinite group G with respect to a closed subset X, say finite, of G: the vertices of Γ are the elements of G, and its directed edges have the form (g, x) (g ∈ G, x ∈ X) with origin d0(g, x) = g and terminal d1(g, x) = gx. Then the topology of G naturally induces a profinite topology on Γ(G,X). Part I of this book contains an exposition of the theory of profinite graphs and how it relates to and is motivated by the theory of profinite groups. Part II deals with applications to profinite groups, while Part III is dedicated to the study of certain properties of abstract groups with the help of tools developed in Parts I and II. Our aim in Parts I and II has been to make the exposition self-contained, and familiarity with the theory of abstract graphs and groups is not strictly necessary. However, knowledge of the BassÐSerre theory certainly helps, and throughout these two parts we often indicate the interconnections. These connections are in fact the main tools for some of the applications to abstract groups in Part III, where results and ideas ranging from topology and abstract group theory to automata theory are used freely. One fundamental difference with the abstract case is that a profinite group acting freely on a profinite tree need not be a free profinite group (it is just projective). This leads to a study of Galois coverings of profinite graphs and fundamental groups of profinite graphs. Throughout the book we have tried to be as general as reasonably possible, and so we consider pro-C groups, where C is a class of finite groups, rather than profinite groups in general. Consequently the book includes studies of Galois C-coverings, C-trees, fundamental groups of graphs of pro-C groups, etc.

vii viii Preface

Part I (Chaps. 2Ð6) includes the development of free products of pro-C groups continuously indexed by a topological profinite space, and a full treatment of the fundamental pro-C group of a graph of pro-C groups. Part II (Chaps. 7Ð10) contains applications to the structure of profinite groups. In Chap. 7 we describe subgroups of fundamental groups of graphs of profinite groups; in particular, an analogue of the Kurosh subgroup theorem for open subgroups of free products of pro-C groups is established. Chapter 8 describes the properties of minimal G-invariant subtrees of a tree on which the group G acts; this is done for profinite as well as abstract groups and graphs. The study of such minimal trees was initiated by Tits when G is cyclic and acts without fixed points on an abstract tree. It turns out that the connections between these types of minimal subtrees in the abstract and profinite cases provides a powerful tool to study certain properties in abstract groups. Chapters 9 and 10 of Part II deal mainly with homology. Chapter 9 includes a theorem of Neukirch and a generalization of Mel’nikov characterizing homologically when a profinite group is the free product of a collection of subgroups continuously indexed by a topological (profinite) space; this plays the role of the usual combinatorial description of free products in the case of abstract groups. This chapter also contains a Kurosh-like theorem for countably generated closed subgroups of free products of pro-p groups due to D. Haran and O. Mel’nikov independently. Chapter 10 includes the well-known theorem of J.-P. Serre that asserts that a torsion-free pro-p group G with an open free pro-p subgroup must be free pro-p. There is also a generalization of this result due to C. Scheiderer, where one allows torsion in G. Using this, the chapter also contains a study of the subgroup of fixed points of an automorphism of a free pro-p group. Part III (Chaps. 11Ð15) contains applications to abstract groups. These include generalizations of a theorem of Marshall Hall that asserts that a finitely generated subgroup H of an abstract free group Φ is the intersection of the subgroups of finite index in Φ that contain H ; an algorithm to compute the closure of a finitely generated subgroup H of an abstract free group Φ in the pro-p topology of Φ; and applications to the theory of formal languages and finite monoids. Also included is the study of certain properties that hold for an abstract group if and only if they hold for the finite quotients of that group, e.g., conjugacy separability for an abstract group R :forx,y ∈ R, these elements are conjugate in R if their images are conjugate in every finite quotient group of R. The book ranges over a large number of areas and results, but we have not in- tended to make this into an encyclopedia of the subject. Part I gives a fairly complete account of profinite graphs and their connection with profinite groups. However in Part II and, even more, in Part III, I have made a choice of topics to illustrate some results and methods. At the end of each of the three parts of the book there is a section with historical comments on the development of the fundamental ideas and theorems, statements of additional results, references to related topics, and open questions. In an effort to make the book self-contained, the first chapter includes a review of basic notions and results about profinite spaces, profinite groups and homology that are used frequently throughout the monograph. Appendix A deals with aspects Preface ix of abstract graphs that are of interest in the book. The main purpose has been to develop a terminology common to abstract and profinite graphs. Appendix B contains a proof of a theorem of M. Benois about rational languages in free abstract groups. I have been indebted to many colleagues during the writing of this book. Throughout the years I have had many mathematical discussions with my longtime collaborator Pavel Zalesskii that have helped to clarify some topics developed here; it is a pleasure to acknowledge with thanks my debt to him. I thank John Dixon, Wolfgang Herfort, Dan Segal and Benjaming Steinberg, who have read parts of the manuscript and have made very useful comments, corrections and suggestions. Jean-Eric Pin has provided helpful references, and I am very grateful to him for this. This book was written mainly in Ottawa and Madrid. In Ottawa my thanks go to Carleton University for continuous help throughout the years, and for sabbat- ical periods that have allowed me to concentrate on the writing of his book. In Madrid I have often used the facilities of the Universidad Complutense, the Uni- versidad Autónoma and ICMAT, and I thank all of them for their generosity, and my colleagues at these institutions for their welcome whenever I have spent time with them. Finally, I acknowledge with thanks the continued research support from NSERC. MadridÐOttawa Luis Ribes March, 2017 Contents

Preface ...... vii 1 Preliminaries ...... 1 1.1 InverseLimits...... 2 1.2 Profinite Spaces ...... 3 1.3 Profinite Groups ...... 4 Pseudovarieties C ...... 5 Generators ...... 6 G-Spaces and Continuous Sections ...... 6 Order of a Profinite Group and Sylow Subgroups ...... 7 1.4 Pro-C Topologies in Abstract Groups ...... 8 1.5 Free Groups ...... 9 1.6 Free and Amalgamated Products of Groups ...... 10 1.7 Profinite Rings and Modules ...... 11 Exact Sequences ...... 13 The Functors Hom(−, −) ...... 13 Projective and Injective Modules ...... 14 1.8 The Complete Group Algebra ...... 15 G-Modules ...... 15 Complete Tensor Products ...... 16 n − − Λ − − 1.9 The Functors ExtΛ( , ) and Torn ( , ) ...... 17 n − − The Functors ExtΛ( , ) ...... 17 Λ − − The Functors Torn ( , ) ...... 19 1.10 Homology and Cohomology of Profinite Groups ...... 20 Cohomology of Profinite Groups ...... 20 Special Maps in Cohomology ...... 23 Homology of Profinite Groups ...... 24 Duality Homology-Cohomology ...... 24 (Co)induced Modules and Shapiro’s Lemma ...... 25 1.11(Co)homologicalDimension...... 25

xi xii Contents

Part I Basic Theory 2 Profinite Graphs ...... 29 2.1 FirstNotionsandExamples...... 29 2.2 Groups Acting on Profinite Graphs ...... 41 2.3 TheChainComplexofaGraph...... 45 2.4 π-Trees and C-Trees...... 48 2.5 Cayley Graphs and C-Trees...... 57 3 The Fundamental Group of a Profinite Graph ...... 63 3.1 GaloisCoverings...... 63 3.2 G(Γ |) as a Subgroup of Aut(Γ ) ...... 72 3.3 Universal Galois Coverings and Fundamental Groups ...... 74 3.4 0-Transversalsand0-Sections...... 77 3.5 Existence of Universal Coverings ...... 82 3.6 Subgroups of Fundamental Groups of Graphs ...... 89 3.7 Universal Coverings and Simple Connectivity ...... 91 3.8 Fundamental Groups and Projective Groups ...... 95 3.9 Fundamental Groups of Quotient Graphs ...... 96 3.10 π-Trees and Simple Connectivity ...... 100 3.11 Free Pro-C Groups and Cayley Graphs ...... 105 3.12 Change of Pseudovariety ...... 107 4 Profinite Groups Acting on C-Trees ...... 111 4.1 FixedPoints...... 111 4.2 Faithful and Irreducible Actions ...... 119 5 Free Products of Pro-C Groups ...... 137 5.1 Free Pro-C Products: The External Viewpoint ...... 137 5.2 Subgroups Continuously Indexed by a Space ...... 145 5.3 Free Pro-C Products: The Internal Viewpoint ...... 148 5.4 Profinite G-Spaces vs the Weight w(G) of G ...... 153 5.5 Basic Properties of Free Pro-C Products ...... 157 5.6 Free Products and Change of Pseudovariety ...... 164 5.7 Constant and Pseudoconstant Sheaves ...... 167 6 Graphs of Pro-C Groups ...... 177 6.1 Graphs of Pro-C Groups and Specializations ...... 177 6.2 The Fundamental Group of a Graph of Pro-C Groups ...... 180 Uniqueness of the Fundamental Group ...... 185 6.3 The Standard Graph of a Graph of Pro-C Groups ...... 193 6.4 Injective Graphs of Pro-C Groups ...... 205 6.5 Abstract vs Profinite Graphs of Groups ...... 207 6.6 Action of a Pro-C Group on a Profinite Graph with Finite Quotient...... 213 6.7 Notes, Comments and Further Reading: Part I ...... 216 Abstract Graph of Finite Groups (G,Γ)over an Infinite Graph Γ 218 Contents xiii

Part II Applications to Proﬁnite Groups

7 Subgroups of Fundamental Groups of Graphs of Groups ...... 223 7.1 Subgroups ...... 223 7.2 Normal Subgroups ...... 227 7.3 The Kurosh Theorem for Free Pro-C Products ...... 232

8 Minimal Subtrees ...... 237 8.1 Minimal Subtrees: The Abstract Case ...... 238 8.2 Minimal Subtrees: Abstract vs Proﬁnite Trees ...... 241 Trees Associated with Virtually Free Groups ...... 242 8.3 Graphs of Residually Finite Groups and the Tits Line ...... 245 8.4 Graph of a Free Product of Groups and the Tits Line ...... 250

9 Homology and Graphs of Pro-C Groups ...... 257 9.1 Direct Sums of Modules and Homology ...... 257 9.2 Corestriction and Continuously Indexed Families of Subgroups 259 9.3 The Homology Sequence of the Action on a Tree ...... 265 9.4 MayerÐVietoris Sequences ...... 267 9.5 Homological Characterization of Free Pro-p Products ..... 270 9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem . . . 272

10 The Virtual Cohomological Dimension of Proﬁnite Groups ..... 279 10.1 Tensor Product of Complexes ...... 279 10.2 Tensor Product Induction for a Complex ...... 281 10.3TheTorsion-FreeCase...... 290 10.4 Groups Virtually of Finite Cohomological Dimension: Periodicity...... 291 10.5TheTorsionCase...... 295 10.6 Pro-p Groups with a Free Subgroup of Index p ...... 309 10.7 Counter Kurosh ...... 312 10.8 Fixed Points of Automorphisms of Free Pro-p Groups ..... 317 10.9 Notes, Comments and Further Reading: Part II ...... 322 M. Hall Pro-p Groups ...... 323

Part III Applications to Abstract Groups

11 Separability Conditions in Free and Polycyclic Groups ...... 329 11.1 Separability Conditions in Abstract Groups ...... 329 11.2 Subgroup Separability in Free-by-Finite Groups ...... 333 11.3 Products of Subgroups in Free Abstract Groups ...... 337 11.4 Separability Properties of Polycyclic Groups ...... 342 xiv Contents

12 Algorithms in Abstract Free Groups and Monoids ...... 349 12.1 Algorithms for Subgroups of Finite Index ...... 349 12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups ...... 353 12.3 Algorithms for Monoids ...... 359 The Kernel of a Finite Monoid ...... 364 The Mal’cev Product of Pseudovarieties of Monoids ...... 366 13 Abstract Groups vs Their Profinite Completions ...... 369 13.1 Free-by-Finite Groups vs Their Profinite Completions ..... 369 13.2 Polycyclic-by-Finite Groups vs Their Profinite Completions . . 379 14 Conjugacy in Free Products and in Free-by-Finite Groups ..... 383 14.1 Conjugacy Separability in Free-by-Finite Groups ...... 383 14.2 Conjugacy Subgroup Separability in Free-by-Finite Groups . . 386 14.3 Conjugacy Distinguishedness in Free-by-Finite Groups ..... 389 15 Conjugacy Separability in Amalgamated Products ...... 391 15.1 Abstract Free Products with Cyclic Amalgamation ...... 392 15.2 Normalizers in Amalgamated Products of Groups ...... 396 15.3 Conjugacy Separability of Amalgamated Products ...... 399 15.4 Amalgamated Products, Quasi-potency and Subgroup Separability ...... 405 15.5 Amalgamated Products and Products of Cyclic Subgroups . . . 407 15.6 Amalgamated Products and Normalizers of Cyclic Subgroups . 411 15.7 Amalgamated Products and Intersections of Cyclic Subgroups . 412 15.8 Amalgamated Products and Conjugacy Distinguishedness . . . 415 15.9 Conjugacy Separability of Certain Iterated Amalgamated Products ...... 418 15.10 Examples of Conjugacy Separable Groups ...... 418 15.11 Notes, Comments and Further Reading: Part III ...... 422 Subgroup Separability and Free Products ...... 423 Conjugacy Separability, Subgroups and Extensions ...... 427 Conjugacy Distinguished Subgroups ...... 427 Appendix A Abstract Graphs ...... 429 A.1 The Fundamental Group of an Abstract Graph ...... 429 TheStarofaVertex...... 430 Paths...... 430 A.2 Coverings of Abstract Graphs ...... 435 A.3 Foldings...... 441 A.4 Algorithms...... 442 Intersection of Finitely Generated Subgroups ...... 443 A.5 Notes, Comments and Further Reading ...... 445 Contents xv

Appendix B Rational Sets in Free Groups and Automata ...... 447 B.1 FiniteStateAutomata:ReviewandNotation...... 447 B.2 The Classical Function ρ ...... 448 B.3 Rational Subsets in Free Groups ...... 449 B.4 Notes, Comments and Further Reading ...... 451 References ...... 453 Index of Symbols ...... 461 Index of Authors ...... 463 Index of Terms ...... 465 Chapter 1 Preliminaries

The purpose of this chapter is to review some notation, terminology, basic concepts and results that are frequently used throughout the book. For more details and proofs one can consult one of the standard references on general proﬁnite groups, e.g., Serre (1994), Wilson (1998), Fried and Jarden (2008), Ribes and Zalesskii (2010). In general we follow the notation in Ribes and Zalesskii (2010), cited as RZ throughout this book.

Basic Notation

Z denotes the group of integers under addition (also the ring of integers). N denotes the set of natural numbers.

H ≤ G, H ≤f G, H ≤o G and H ≤c G indicate that H is a subgroup of the group G, respectively, of ﬁnite index, open, closed.

H G, H f G, H o G, H c G indicate that H is a normal subgroup of the group G, respectively, of ﬁnite index, open, closed.

S ⊆o T and S ⊆c T indicate that S is an open, respectively closed, subset of the topological space T . If H and K are subgroups of a group R, then −1 NK (H ) = x ∈ K x Hx = H denotes the normalizer of H in K, and −1 CK (H ) = x ∈ K x hx = h, ∀h ∈ H denotes the centralizer of H in K. Let x and y be elements of a group R. Then xy = y−1xy and yx = yxy−1.

© Springer International Publishing AG 2017 1 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_1 2 1 Preliminaries

Cn denotes the cyclic group of order n, written multiplicatively.

Fp denotes the prime ﬁeld with p elements (occasionally also the additive group of that ﬁeld). For other pieces of notation the reader may consult the Index.

1.1 Inverse Limits

A poset (or partially ordered set) (I, ) is said to be directed if whenever i, j ∈ I , there exists some k ∈ I such that i, j k.Aninverse system of topological spaces (respectively, topological groups) over such a directed poset (I, ) consists of a collection

{Xi | i ∈ I} of topological spaces (respectively, topological groups) indexed by I , and a collection of continuous mappings (respectively, continuous group homomorphisms) ϕij : Xi −→ Xj , deﬁned whenever i j, such that the diagrams of the form

ϕik Xi Xk

ϕij ϕjk

Xj commute whenever they are deﬁned, i.e., whenever i, j, k ∈ I and i j k.LetX be a topological space (respectively, topological group) and let

ϕi : X −→ Xi (i ∈ I) be a collection of continuous mappings (respectively, continuous group homomorphisms) that are compatible (i.e., ϕij ϕi = ϕj whenever j i). One says that X, together with these mappings, is an inverse limit of the inverse system {Xi,ϕij ,I} if the following universal property is satisﬁed:

ψ Y X

ϕi ψi Xi whenever Y is a topological space (respectively, topological group) and ψi : Y −→ Xi (i ∈ I) is a set of compatible continuous mappings (respectively, continuous homomorphisms), then there is a unique continuous mapping (respectively, continuous homomorphism) ψ : Y −→ X such that ϕiψ = ψi , for all i ∈ I .The maps ϕi : X −→ Xi are called projections. It follows easily from the definition that inverse limits are unique, if they exist. An inverse limit X of an inverse system {Xi,ϕij ,I} of topological spaces (respectively, topological groups) over a directed poset I can be constructed as follows: define X 1.2 Profinite Spaces 3 as the subspace (respectively, subgroup) of the direct product i∈I Xi of topological spaces (respectively, topological groups) consisting of those tuples (xi) that satisfy the condition ϕij (xi) = xj ,ifi j.Let

ϕi : X −→ Xi −→ denote the restriction of the canonical projection i∈I Xi Xi . Then one easily checks that each ϕi is continuous (respectively, a continuous homomorphism), and that (X, ϕi) is an inverse limit. These definitions of inverse systems and inverse limits translate in an obvious manner to other categories: one can replace topological spaces and continuous functions with sets and maps of sets, rings and homomorphisms of rings, etc. Next we state some useful facts that we use often in this book. The first one says that the inverse limit of an inverse system of nonempty finite sets is nonempty, or in more generality:

1.1.1 Let {Xi,ϕij ,I} be an inverse system of compact Hausdorff nonempty topological spaces Xi over the directed poset (I, ). Then

lim←− Xi i∈I is nonempty.

Let (I, ) be a directed poset. Assume that I is a subset of I in such a way that (I , ) becomes a directed poset. We say that I is cofinal in I if for every i ∈ I there is some i ∈ I such that i i .If{Xi,ϕij ,I} is an inverse system and I is cofinal in I , then {Xi,ϕij ,I } becomes an inverse system in an obvious way, and we say that {Xi,ϕij ,I } is a cofinal subsystem of {Xi,ϕij ,I}.

1.1.2 Let {Xi,ϕij ,I} be an inverse system of compact topological spaces (respectively, compact topological groups) over a directed poset I and assume that I is a coﬁnal subset of I . Then

=∼ lim←− Xi lim←− Xi . i∈I i ∈I

1.2 Proﬁnite Spaces

A profinite space (sometimes called a boolean space) X is an inverse limit = X lim←− Xi (1.1) i∈I of an inverse system of finite spaces (endowed with the discrete topology) {Xi,ϕij ,I}. Recall that a topological space is totally disconnected if every point in the space is its own connected component. One can then describe a profinite space in terms of internal topological properties as follows: 4 1 Preliminaries

1.2.1 A topological space is proﬁnite if and only if it is compact, Hausdorff and totally disconnected.

In one direction, this is a consequence of the construction of the inverse limit as a subspace of the direct product of ﬁnite spaces and Tychonoff’s theorem, which asserts that the direct product of compact spaces is compact. Another important property of proﬁnite spaces that is very useful is that its topology is determined by its closed and open (or clopen) subsets:

1.2.2 A topological space is proﬁnite if and only if it is compact, Hausdorff and its topology admits a base of clopen sets.

In (1.1) denote by ϕi : X → Xi the projection into Xi . Then one has

1.2.3 If Y ⊆ X, then the topological closure Y¯ of Y in the proﬁnite space X is the inverse limit of the images ϕi(Y ). Consequently, two subspaces of X with the same images under every ϕi have the same closure.

Let X be a topological space. Deﬁne the weight w(X) of X to be the smallest cardinal of a base of open sets of X. We denote by ρ(X) the cardinal of the set of all clopen subsets of X. A topological space X is said to be second countable,orto satisfy the second axiom of countability, if it has a countable base of open sets.

1.2.4 If X is an inﬁnite proﬁnite space, then w(X) = ρ(X). In particular, the cardinality of any base of open sets of X consisting of clopen sets is ρ(X).

1.2.5 A proﬁnite space X is second countable if and only if = X lim←− Xi, i∈I where (I, ) is a countable totally ordered set and each Xi is a ﬁnite discrete space.

1.3 Proﬁnite Groups

A profinite group G is an inverse limit of finite groups Gi = G lim←− Gi, (1.2) i∈I and so one checks that the class of profinite groups is exactly the class of those topological groups that are compact, Hausdorff and totally disconnected. Equivalently, profinite groups are precisely Galois groups of Galois extensions of fields. Other concrete examples of profinite groups arise (and this is of importance in Part III of this book) when taking completions of abstract groups endowed with certain natural topologies (see Sect. 1.4 below). 1.3 Profinite Groups 5

Pseudovarieties C

Sometimes it is convenient to be more precise and consider finite groups Gi in the above inverse limit with specific properties. For this purpose one introduces the notion of a pseudovariety of finite groups: a nonempty class of finite groups C is a pseudovariety1 if it is closed under taking subgroups, homomorphic images and finite direct products. Important examples of pseudovarieties of finite groups: the class of all finite groups; the class of all finite solvable groups; the class of all finite nilpotent groups; the class of all finite p-groups (where p is a fixed prime number); the class of all finite abelian groups, etc. Let C be a fixed pseudovariety of finite groups. A pro-C group G is an inverse limit (1.2) of finite groups Gi in C. Thus one speaks of a prosolvable group, a pronilpotent group, a pro-p group or a proabelian group, depending on which pseudovariety C, among those mentioned above, has been selected. A pseudovariety of finite groups C is said to be extension-closed if whenever 1 −→ K −→ G −→ H −→ 1 is an exact sequence of finite groups with K,H ∈ C, then G ∈ C. The pseudovariety C is said to be closed under extensions with abelian kernel if whenever 1 −→ K −→ G −→ H −→ 1 is an exact sequence of finite groups with K,H ∈ C and K is abelian, then G ∈ C. For a pseudovariety of finite groups C, π(C) denotes the set of all prime numbers p that divide the order of some group G in C. We also say that π(C) is the set of primes involved in C. Of course, every pro-C group is automatically profinite. The topology of a profinite group is determined by its open normal subgroups, and one has the following useful way of describing pro-C groups:

1.3.1 A group G is pro-C if and only if G is a topological group with a fundamental system U of open neighbourhoods U of the identity element 1 such that each U is a normal subgroup of UG with G/ ∈ C, and = G lim←− G/U. U∈U

The following property is often useful.

1.3.2 Let G be a profinite group and let (1.2) be a decomposition of G as an inverse limit of finite groups Gi . Then every continuous homomorphism ϕ : G → H from G to a finite group H factors through one of the groups Gi ; i.e., there is some i ∈ I and some homomorphism ρ : Gi → H such that ϕ = ρϕi , where ϕi : G → Gi is the canonical projection.

A similar result holds for proﬁnite spaces.

1In some publications, including RZ, this is called a ‘variety’ of ﬁnite groups. 6 1 Preliminaries

Generators

Let G be a profinite group and let X ⊆ G. One says that X generates G (as a topological group) if the abstract subgroup X of G generated by X is dense in G. We write G = X. We say that a subset X of a profinite group G converges to 1 if every open subgroup U of G contains all but a finite number of the elements in X.IfX generates G and converges to 1, then we say that X is a set of generators of G converging to 1. This definition is motivated by the following fact.

1.3.3 Every proﬁnite group G admits a set of generators converging to 1.

If G is a proﬁnite group, d(G) denotes the smallest cardinality of a set of generators of G converging to 1. It turns out that the cardinal d(G) (or more precisely, whether or not this cardinal is at most ℵ0) plays an important role concerning the structure of certain groups acting on proﬁnite trees, as we shall see later on in this book; and one reason for this is that some properties of such groups can be proved using induction, using the following characterization.

1.3.4 Let C be a pseudovariety of ﬁnite groups and let G be a pro-C group. Then d(G) ≤ℵ0 if and only if G has a countable descending chain of open normal subgroups

G = G0 ≥ G1 ≥···≥Gi ≥··· ∞ = such that i=0 Gi 1.

Finitely generated proﬁnite groups are in addition Hopﬁan, that is, they satisfy the following useful property.

1.3.5 Let G be a ﬁnitely generated proﬁnite group. Then every continuous epimorphism ϕ : G −→ G is an isomorphism.

G-Spaces and Continuous Sections

Let G be a profinite group. We say that a profinite space X is a left G-space,or simply a G-space,ifG acts continuously on X, that is, if there exists a continuous map G × X → X, denoted (g, x) → gx, such that (gh)x = g(hx) and 1x = x, for all g,h ∈ G, x ∈ X. A pointed topological space (X, ∗) is a G-space if X is a G-space in the above 1.3 Profinite Groups 7 sense, and in addition g∗=∗for all g ∈ G. The action of G on X (respectively (X, ∗))issaidtobefree if for every x ∈ X (respectively, for every x ∈ X −{∗}) and every 1 = g ∈ G one has gx = x.

1.3.6 Let G be a proﬁnite group acting on a proﬁnite space X. Then

(a) X admits a decomposition as an inverse limit of finite G-spaces Xi : = X lim←− Xi. i∈I (b) Suppose that G is finite and acts freely on X. Then X admits a decomposition as an inverse limit of finite free G-spaces Xi : = X lim←− Xi. i∈I

Similar results hold for pointed proﬁnite G-spaces (X.∗). Associated with a G-space X there a quotient space G\X, whose elements are the G-orbits Gx (x ∈ X), which in turn is a proﬁnite space if one endows G\X with the quotient topology. A section of the natural continuous epimorphism ϕ : X −→ G\X (1.3) is a map σ : G\X → X such that ϕσ = idG\X. In general, quotient maps of the form (1.3) do not admit continuous sections. In fact

1.3.7 For any nontrivial proﬁnite group G, there is a proﬁnite G-space X such that the natural continuous epimorphism ϕ : X −→ G\X does not admit a continuous section.

But the existence of continuous sections is guaranteed in the following important cases.

1.3.8 If H is a closed subgroup of a proﬁnite group G, then the quotient map ϕ : G −→ H \G of G onto the space of right cosets of H admits a continuous section.

1.3.9 If a proﬁnite group G acts freely on a proﬁnite space X, then the quotient map ϕ : X −→ G\X admits a continuous section.

1.3.10 If a proﬁnite group G acts on a second countable proﬁnite space X, then the quotient map ϕ : X −→ G\X admits a continuous section.

Order of a Proﬁnite Group and Sylow Subgroups

If G is an infinite profinite group, knowing its cardinality provides little information. There is, nevertheless, a very useful notion of order of a profinite group G that 8 1 Preliminaries reflects, in a global manner, the arithmetic properties of the finite quotient groups G/U , where U is an open normal subgroup of G. To each prime number p one associates an exponent n(p) (a natural number or the symbol ∞) that is the supre- mum (over all open normal subgroups of G) of all the natural numbers n such that pn divides the order of the finite group G/U . Then the order #G of the profinite group G is the formal product #G = pn(p) p (‘a supernatural number’). If p is a fixed prime number, a pro-p group has order pn, where 0 ≤ n ≤∞.Ap-Sylow subgroup P of a profinite group G is a maximal pro-p subgroup of G. Using Zorn’s Lemma one sees that G contains p-Sylow subgroups. They have properties analogous to those of the p-Sylow subgroups of finite groups: any two p-Sylow subgroups of G are conjugate and any pro-p subgroup of G is contained in one of its p-Sylow subgroups.

1.4 Pro-C Topologies in Abstract Groups

Let C be a pseudovariety of ﬁnite groups and let R be an abstract group. Deﬁne

N = NC(R) ={N R | R/N ∈ C}. One can make R into a topological group by taking N as a base of open neighbourhoods of the identity element 1 of R; this topology is called the pro-C topology of R or, if emphasis is needed, the full pro-C topology of R. Note that the pro-C topology of R is Hausdorff if and only if N = 1. N∈N If that is the case, R is said to be residually C. When C is the pseudovariety of all finite groups (respectively, all finite p-groups, all finite solvable groups, etc.), one uses instead the expression residually finite (respectively, residually p, residually solvable,etc.). The pro-C completion of R is defined to be the pro-C group = RCˆ lim←− R/N. N∈N

The canonical homomorphism of R into RCˆ

ι : R −→ RCˆ is given by r → (rN)N∈N (r ∈ R). Note that ι is injective if and only if R is residually C; ι(R) is a dense subgroup of RCˆ.IfC is the pseudovariety of all finite groups, ˆ one usually writes the completion as R, instead of RCˆ, and calls it the profinite completion of R.IfC is the pseudovariety of all finite p-groups, where p is a fixed prime number, then one usually writes the completion as Rpˆ . 1.5 Free Groups 9

For example, the proﬁnite completion of the group Z is denoted by Zˆ : ˆ = Z lim←− Z/nZ. n∈N

Its pro-p completion is denoted by Zp (rather than Zpˆ ), following a long tradition in number theory: = n Zp lim←− Z/p Z. n∈N ˆ Note that Z is naturally embedded in Z and in Zp. Observe also that one can think ˆ ˆ of Zp as being embedded in Z; in fact Zp is the unique p-Sylow subgroup of Z. Moreover, Zˆ is the direct product of its Sylow subgroups. An additional useful ob- ˆ servation is that Z and Zp naturally have a ring structure (they are examples of proﬁnite rings). One usually refers to Zp as the ring of p-adic integers.

1.5 Free Groups

Let C be a pseudovariety of ﬁnite groups and let X be a ﬁnite set. Denote by Φ the free abstract groups with basis X. Then the pro-C completion

F = F(X)= ΦCˆ of Φ is called the free pro-C group on the finite set X (a basis of the pro-C group). It satisfies the expected universal property of a free object in the category of pro-C groups: if we denote by ι : X → F the natural map (it is an injection in general), then for every function ϕ : X → H into a pro-C group, there exists a unique continuous homomorphism ϕ¯ : F → H such that ϕι¯ = ϕ. One can define a free pro-C group with more general bases, namely when X is a profinite space (or a pointed profinite space). The universal property that defines such a free pro-C group F = F(X)is analogous to the universal property described above, one simply requires that ϕ be continuous (and a map of pointed spaces, if (X, ∗) is a pointed space). Again, one may assume that X is a subspace of F .One important peculiarity is that a free pro-C group on a pointed profinite space (X, ∗) is also a free pro-C group on a pointed space which is the one-point compactification of a set (a discrete space) S; then one can think of S as a set of generators of F converging to 1. The cardinality of such S is an invariant of the group F ; this allows us to define the rank of the free pro-C group F : rank(F ) =|S|. ˆ For example, Z is the free profinite group of rank 1, while Zp is the free pro-p group of rank 1.

1.5.1 If G is a pro-C group, there exists a free pro-C group F and a continuous epimorphism F → G. 10 1 Preliminaries

Closed subgroups of free pro-C groups need not be free pro-C (e.g., a p-Sylow subgroup of a free proﬁnite group).

1.5.2 Open subgroups of free pro-C groups are free pro-C groups (if C is extension- closed).

1.5.3 Closed subgroups of free pro-p groups are free pro-p groups.

A proﬁnite group P is called projective if it satisﬁes the following universal property: P ϕ¯ ϕ

H α G whenever H and G are proﬁnite groups, ϕ : P → G is a continuous homomorphism and α : H → G is a continuous epimorphism, there exists a continuous homomorphism ϕ¯ : P → H such that ϕ = αϕ¯.

1.5.4 Projective proﬁnite groups are precisely the closed subgroups of free proﬁnite groups.

1.6 Free and Amalgamated Products of Groups

A free pro-C product C G = G1 G2 = G1 G2 of two pro-C groups G1 and G2 is a pro-C group G together with continuous homomorphisms ϕ1 : G1 −→ G and ϕ2 : G2 −→ G satisfying the following universal property: G ψ ϕi

Gi L ψi for any pro-C group L and any continuous homomorphisms ψi : Gi −→ L (i = 1, 2), there is a unique continuous homomorphism ψ : G −→ L such that ψi = ψϕi (i = 1, 2). From this definition one easily deduces that the homomorphisms ϕ1 and ϕ2 are injective and that the free pro-C product G is unique up to isomorphism. One constructs G as a completion of the free product G1 ∗ G2 of G1 and G2 as abstract groups, with respect to a certain pro-C topology on G1 ∗ G2 that ensures that ϕ1 and ϕ2 are continuous. For example, the free profinite group Fn of finite rank n is the free profinite product of n copies of Zˆ . 1.7 Profinite Rings and Modules 11

Next we remind the reader about free products with amalgamation (also called ‘amalgamated products’ or ‘amalgamated free products’) in the category of profinite groups (i.e., when C is the pseudovariety of all profinite groups). Let G1, G2 and K be profinite groups. Let fi : K −→ Gi (i = 1, 2) be continuous monomorphisms (here we think of them as inclusion maps). An amalgamated free profinite product of G1 and G2 with amalgamated subgroup K is defined to be a profinite group, denoted

G = G1 K G2, together with continuous homomorphisms

ϕi : Gi −→ G, (i = 1, 2), with ϕ1f1 = ϕ2f2, satisfying the following universal property: for any pair of continuous homomorphisms ψ1 : G1 −→ L, ψ2 : G2 −→ L into a proﬁnite group L with ψ1f1 = ψ2f2, there exists a unique continuous homomorphism ψ : G −→ L such that the following diagram is commutative

f1 K G1

f2 ϕ1

ϕ2 ψ1 G2 G ψ

ψ2 L Again the uniqueness of G is easily deduced from the definition. One constructs G as the completion of the free product G1 ∗K G2 with amalgamation of G1 and G2 amalgamating K as abstract groups, with respect to a certain profinite topology on G1 ∗K G2 to ensure that ϕ1 and ϕ2 are continuous. In this case it is not automatic that the homomorphisms ϕ1 and ϕ2 are injective; the main reason for this is that, in general, the abstract group G1 ∗K G2 is not residually finite. When ϕ1 and ϕ2 are injective, one says that the amalgamated free profinite product G1 K G2 is proper. Some proper amalgamated free profinite products are studied in detail in this book; this is an important prerequisite for using methods developed in this book to study certain properties of abstract groups of the form R1 ∗H R2.

1.7 Proﬁnite Rings and Modules

A profinite ring Λ is an inverse limit of an inverse system {Λi,ϕij ,I} of finite rings. We always assume that rings have an identity element, usually denoted by 1, and that homomorphisms of rings send identity elements to identity elements. ˆ As pointed out earlier, Z and Zp, and more generally ZCˆ, are examples of profinite rings. Their quotient rings will also play an important role in parts of this book. 12 1 Preliminaries

We are interested in two types of Λ-modules M (‘Λ-module’ is meant in the usual abstract sense, but in addition we assume that M is a topological abelian group and that the action of Λ on M is continuous), namely those with the discrete topology and those with a profinite topology. In general, we assume that the action of Λ on M is on the left; for emphasis sometimes we write ‘left Λ-module’. If the action of Λ on M is on the right, we write ‘right Λ-module’. We will need both left and right Λ-modules in the sequel. The category of discrete Λ-modules and their Λ-homomorphisms is denoted by DMod(Λ). The category of profinite Λ-modules and their continuous Λ- homomorphisms is denoted by PMod(Λ). A profinite ring Λ has a fundamental system for the neighbourhoods of 0 consisting of open (two-sided) ideals; from this one deduces the following fact for discrete modules.

1.7.1 If M is a discrete Λ-module, then M is the union of its ﬁnite Λ-submodules; in particular, M is torsion as an abelian group.

Proﬁnite Λ-modules have properties similar to those of proﬁnite groups. For example,

1.7.2 If M is a proﬁnite Λ-module, then M is the inverse limit of its ﬁnite quotient Λ-modules.

In analogy with the definition of a free profinite group, one defines a free profinite Λ-module on a pointed profinite space (X, ∗) to consist of a profinite Λ-module, denoted [[ Λ(X, ∗)]] , together with a map of pointed spaces

ι : (X, ∗) −→ Λ(X, ∗) (i.e., ι(∗) = 0) satisfying the following universal property: whenever ϕ : (X, ∗) −→ N is a continuous mapping of pointed spaces into a profinite Λ-module N, there exists a unique continuous Λ-homomorphism ϕ¯ :[[Λ(X, ∗)]] −→ N such that ϕι¯ = ϕ. One has a similar definition for a free profinite Λ-module, denoted [[ ΛX]] ,on a profinite space X.IfX is a set and R is any ring, we denote the free abstract R-module with basis X by [RX] (it is the direct sum of |X| copies of R, considered as an abstract R-module in a natural way). When X is finite, [[ ΛX]] = [ ΛX] (in this case we understand the topology on [ΛX] to be the product topology). As in the case of profinite groups, one develops the concepts of ‘set of generators converging to 0’ in a Λ-module and of ‘free profinite Λ-module on a set converging to 0’. There are results similar to 1.3.3 and 1.5.1: 1.7 Profinite Rings and Modules 13

1.7.3 Every profinite Λ-module admits a set of generators converging to 0 and every profinite Λ-module is the image of a continuous Λ-homomorphism from a free profinite Λ-module.

Exact Sequences

Let ϕi−1 ϕi ···−→Mi−1 −→ Mi −→ Mi+1 −→··· be a sequence of Λ-modules and Λ-homomorphisms. This sequence is said to be exact at Mi if Im(ϕi−1) = Ker(ϕi). If this is the case at each module of the sequence, we say that the sequence is exact. Sequences of the form 0 −→ A −→ B −→ C −→ 0, where 0 denotes the module consisting only of the element zero, are used frequently. When they are exact, one refers to them as ‘short exact sequences’.

The Functors Hom(−, −)

If M and N are Λ-modules, we denote by HomΛ(M, N) the abelian group of continuous homomorphisms from M to N.IfweﬁxM, then we denote by HomΛ(M, −) the function that assigns to the Λ-module N the abelian group HomΛ(M, N). This function behaves ‘functorially’ (in fact, we refer to HomΛ(M, −) as a functor, or more speciﬁcally as a ‘covariant’ functor) in the sense that a continuous Λ-homomorphism ϕ1 : N1 → N2 of Λ-modules determines a homomorphism of abelian groups

HomΛ(M, ϕ1) : HomΛ(M, N1) −→ HomΛ(M, N2) deﬁned by

HomΛ(M, ϕ1)(ψ) = ϕ1ψ ψ ∈ HomΛ(M, N1) ; and furthermore, (1) if ϕ2 : N2 → N3 is also a continuous Λ-homomorphism of Λ-modules, then HomΛ(M, ϕ2ϕ1) = HomΛ(M, ϕ2)HomΛ(M, ϕ1), and (2) if idN : N → N is the identity homomorphism, then HomΛ(M, idN ) : HomΛ(M, N) → HomΛ(M, N) is the identity homomorphism. A functor that sends exact sequences to exact sequences is called an exact functor (to check whether this is the case, it is enough to consider short exact sequences). The functor HomΛ(M, −) is not exact in general, but it is always left exact, meaning that if 0 −→ A −→ B −→ C −→ 0 14 1 Preliminaries is a short exact sequence of Λ-modules and continuous Λ-homomorphisms, then the sequence of abelian groups

0 −→ HomΛ(M, A) −→ HomΛ(M, B) −→ HomΛ(M, C) is also exact. Similarly, Hom(−,N) is also a functor (it is called ‘contravariant’ because it reverses the arrows). In general it is not exact, but it is always left exact.

Projective and Injective Modules

A profinite Λ-module P is called projective (in the category of profinite Λ-modules PMod(Λ)) if is a direct summand of a free profinite Λ-module of the form [[ ΛX]] , i.e., [[ ΛX]] = P ⊕ P , where P is some profinite Λ-submodule of [[ ΛX]] .Inview of 1.7.3, this is equivalent to saying that P satisfies the following universal property: P ϕ¯ ϕ

B α A whenever ϕ : P → A is a continuous Λ-homomorphism and α : B → A is a continuous Λ-epimorphism of profinite Λ-modules, there exists a continuous Λ- homomorphism ϕ¯ : P → B such that ϕ = αϕ¯. This second definition is ‘better’ in the sense that it can be formulated in any category (not just for profinite modules). Still another equivalent way of indicating that P is projective is to say that the functor HomΛ(P, −) is exact. In particular, it follows from the first definition that every free profinite Λ-module is projective and so (see 1.7.3) every profinite Λ-module is the homomorphic image of a projective module in the category PMod(Λ). One sometimes expresses this by saying that the category PMod(Λ) ‘has enough projectives’. This is important when developing a homology theory for profinite modules, as we will see in Sect. 1.9. Another important observation is that to check whether or not a profinite Λ-module P is projective, one may assume that in the diagram above the modules A and B are finite. The dual concept of ‘projective object’ is ‘injective object’. We recall the definition explicitly for discrete Λ-modules (i.e., in the category DMod(Λ)). One says that a discrete Λ-module Q is injective if whenever α : A → B is a Λ- monomorphism of discrete Λ-modules and ϕ : A → Q is a Λ-homomorphism,

A α B

ϕ ϕ¯ Q 1.8 The Complete Group Algebra 15 then there exists a Λ-homomorphism ϕ¯ : B −→ Q making the diagram commutative, i.e., ϕα¯ = ϕ. Equivalently, Q is injective in DMod(Λ) if the functor Hom(−,Q)is exact, i.e., whenever 0 −→ A −→ B −→ C −→ 0 is an exact sequence in DMod(Λ), so is the corresponding sequence 0 −→ Hom(C, Q) −→ Hom(B, Q) −→ Hom(A, Q) −→ 0 of abelian groups.

1.7.4 Every discrete Λ-module can be embedded as a submodule of an injective discrete Λ-module, i.e., DMod(Λ) ‘has enough injectives’.

1.8 The Complete Group Algebra

Consider a commutative profinite ring R and a profinite group H . We denote the usual abstract group algebra (or group ring) by [RH]. Recall that it consists of all ∈ formal sums h∈H rhh (rh R, where rh is zero for all but a finite number of indices h ∈ H ), with natural addition and multiplication. As an abstract R-module, [RH] isfreeonthesetH . ∼ Assume that H is a finite group. Then [RH] is (as a set) a direct product [RH] = | | [ ] [ ] H R of H copies of R.Ifweimposeon RH the product topology, then RH becomes a topological ring, in fact a profinite ring (since this topology is compact, Hausdorff and totally disconnected). Suppose now that G is a profinite group. Define the complete group algebra (or complete group ring) [[ RG]] to be the inverse limit

[[ ]] = RG lim←− R(G/U) U∈U of the ordinary group algebras [R(G/U)], where U is the collection of all open normal subgroups of G. One can express [[ RG]] as an inverse limit of ﬁnite rings [(R/I)(G/U)], where I and U range over the open ideals of R and the open normal subgroups of G, respectively.

G-Modules

If G is a proﬁnite group, a left G-module consists of a topological abelian group M together with a continuous map G × M → M, denoted by (g, a) → ga, satisfying the following conditions: (i) (gh)m = g(hm); (ii) g(m + m ) = gm + gm ; (iii) 1m = m, for all g,h ∈ G, m, m ∈ M, where 1 is the identity of G. 16 1 Preliminaries

For example, if N/K is a Galois extension of fields and G is its Galois group, then N + (the additive group of N) and the group of roots of unity in N (under multiplication) are examples of (discrete) G-modules. Every [[ RG]] -module is naturally a G-module. It is easy to see that a profinite abelian group naturally has the structure of a Zˆ -module, and so one has that a profinite G-module is the same as a profinite [[ Zˆ G]] -module.

1.8.1 Denote by DMod(G) the category of all discrete G-modules and G- homomorphisms; then DMod([[ZG]] ) coincides with the subcategory of DMod(G) consisting of the discrete torsion G-modules.

1.8.2 DMod(G) has enough injectives.

Complete Tensor Products

ˆ Denote by R a commutative profinite ring, for example, a quotient of Z: Zp (for some prime number p), or ZCˆ (for some pseudovariety of finite groups C), or Fp (the finite prime field with p elements). Let G be a profinite group and let Λ =[[RG]] be the corresponding complete group algebra. Let A be a profinite right Λ-module, B a profinite left Λ-module, and let M be an R-module. A continuous map ϕ : A × B −→ M is called middle linear if ϕ(a + a ,b)= ϕ(a,b)+ ϕ(a ,b), ϕ(a,b+ b ) = ϕ(a,b)+ ϕ(a,b ) and ϕ(ar,b) = ϕ(a,rb), for all a,a ∈ A, b,b ∈ B, r ∈ R. We say that a profinite R-module T together with a continuous middle linear map A × B −→ T , denoted (a, b) → a ⊗ b,isacomplete tensor product of A and B over Λ if the following universal property is satisfied: If M is a profinite R-module and ϕ : A × B −→ M a continuous middle linear map, then there exists a unique map of R-modules ϕ¯ : T −→ M such that ϕ(a¯ ⊗ b) = ϕ(a,b). The complete tensor product of A and B is unique, up to isomorphism, and it is denoted by T = A ⊗Λ B. It can be expressed in terms of tensor products of finite abstract modules:

1.8.3 If one writes A as an inverse limit of ﬁnite right Λ-modules Ai , and B as an inverse limit of ﬁnite left Λ-modules Bi , then A ⊗ B = lim(A ⊗ B ), Λ ←− i Λ j where Ai ⊗Λ Bj is the usual tensor product as abstract Λ-modules. n − − Λ − − 1.9 The Functors ExtΛ( , ) and Torn ( , ) 17

It is useful to recall some basic properties of complete tensor products; they allow us to work with ease without always having to appeal to the definition. 1.8.4 A ⊗Λ − is an additive functor, i.e., it sends finite direct sums to finite direct sums: A ⊗Λ (B1 ⊕ B2) = A ⊗Λ B1 ⊕ A ⊗Λ B2. 1.8.5 A ⊗Λ − is a right exact functor: if

0 → B1 → B2 → B3 → 0 is a short exact sequence of left Λ-modules, then the corresponding sequence of R-modules A ⊗Λ B1 −→ A ⊗Λ B2 −→ A ⊗Λ B3 −→ 0 is exact. 1.8.6 If P is a projective right Λ-module, then the functor P ⊗Λ − is exact. There are similar statements for the functor − ⊗ΛB.

n − − Λ − − 1.9 The Functors ExtΛ( , ) and Torn ( , ) We restrict ourselves to profinite rings of the form Λ =[[RG]] (complete group rings), where R is a commutative profinite ring and G is a profinite group, although most concepts can be described for more general profinite R-algebras Λ in a similar manner.

n − − The Functors ExtΛ( , )

Let A be a proﬁnite Λ-module and B a discrete Λ-module. For each natural number = n n 0, 1,..., one deﬁnes a discrete R-module ExtΛ(A, B) as follows. Consider an exact sequence (called a projective resolution of A)

···−→Pn+1 −→ Pn −→···−→P0 −→ A −→ 0 (1.4) where each Pi is a projective Λ-module; such a projective resolution exists because PMod(Λ) has enough projectives. Applying the functor Hom(−,B) we obtain a sequence

fn 0 −→ HomΛ(P0,B)−→···−→HomΛ(Pn,B)−→ HomΛ(Pn+1,B)−→··· n which is not exact in general. Then ExtΛ(A, B) is deﬁned to be the n-th cohomology group of this sequence: n = ExtΛ(A, B) Ker(fn)/Im(fn−1). 18 1 Preliminaries

n − It turns out that ExtΛ(A, B) can also be computed using the functor HomΛ(A, ): consider an injective resolution of B + 0 −→ B −→ Q0 −→···−→Qn −→ Qn 1 −→···

(i.e., this is an exact sequence where each Qi is a discrete injective Λ-module; it exists because DMod(Λ) has enough injectives). Apply HomΛ(A, −) to obtain n 0 n g n+1 0 −→ HomΛ A,Q −→···−→HomΛ A,Q −→ HomΛ A,Q −→··· This sequence is not exact in general and one has

n = n n−1 ExtΛ(A, B) Ker g /Im g . It is rather cumbersome to work directly with these computational deﬁnitions. The following properties provide enough information for most purposes. We just write n − n − them for ExtΛ(A, ) (there are similar properties for ExtΛ( ,B)).

0 = 1.9.1 ExtΛ(A, B) HomΛ(A, B). n = ≥ 1.9.2 If B is an injective discrete Λ-module, then ExtΛ(A, B) 0, for n 1. n − 1.9.3 For each n,ExtΛ(A, ) is an additive functor.

1.9.4 If 0 → B1 → B2 → B3 → 0 is a short exact sequence of discrete Λ-modules, then there exist ‘connecting homomorphisms’ = n : n −→ n+1 δ δ ExtΛ(A, B3) ExtΛ (A, B1) (n = 0, 1,...)satisfying the following conditions: (a) For every commutative diagram

0 B1 B2 B3 0

0 B1 B2 B3 0 of discrete Λ-modules with exact rows, the following diagram commutes for every n

n δ n+1 ExtΛ(A, B3) ExtΛ (A, B1)

n δ n+1 ExtΛ(A, B3) ExtΛ (A, B1) (b) The long sequence ···→ n−1 →δ n → n → n →···δ ExtΛ (A, B3) ExtΛ(A, B1) ExtΛ(A, B2) ExtΛ(A, B3) is exact. n − − Λ − − 1.9 The Functors ExtΛ( , ) and Torn ( , ) 19

{ n − } = The above properties of the sequence of functors ExtΛ(A, ) (n 0, 1,...) are sufﬁcient to characterize them, in a sense that one can make precise; it says, for example, that any other sequence of functors of the same type satisfying these prop- { n − } = erties must coincide with ExtΛ(A, ) (n 0, 1,...). One refers to such sequences as being universal. This is often very useful because to understand the functors n − 0 − ExtΛ(A, ) in every dimension n, it sufﬁces to study the behaviour of ExtΛ(A, ).

Λ − − The Functors Torn ( , )

Let A be a profinite right Λ-module and B a profinite left Λ-module. For n = Λ 0, 1,..., one defines a profinite R-module Torn (A, B) as follows. Assume that (1.4) Λ is a projective resolution of the profinite right Λ-module A. Then Torn (A, B) is the n-th homology group of the sequence

fn ···−→Pn+1 ⊗ΛB −→ Pn ⊗ΛB −→···−→P0 ⊗ΛB −→ 0, i.e., Λ = Torn (A, B) Ker(fn−1)/Im(fn). Λ ⊗ − It turns out that one can also calculate Torn (A, B) using the functor A Λ :itis the n-th homology group of the sequence obtained by applying A ⊗Λ− to a projec- n − − tive resolution of B. As in the case of ExtΛ( , ), this computational deﬁnition of Λ Torn (A, B) is hard to use; instead one relies on some basic properties that we list next.

Λ = ⊗ 1.9.5 Tor0 (A, B) A ΛB.

Λ = ≥ 1.9.6 If B is a projective proﬁnite left Λ-module, then Torn (A, B) 0, for n 1.

Λ − 1.9.7 For each n,Torn (A, ) is an additive functor.

1.9.8 If 0 → B1 → B2 → B3 → 0 is a short exact sequence of proﬁnite left Λ- modules, then there exist ‘connecting homomorphisms’ = : Λ −→ Λ δ δn Torn+1(A, B3) Torn (A, B1) (n = 0, 1,...)satisfying the following conditions: (a) For every commutative diagram

0 B1 B2 B3 0

0 B1 B2 B3 0 20 1 Preliminaries

of proﬁnite left Λ-modules with exact rows, the following diagram commutes for every n

Λ δ Λ Torn+1(A, B3) Torn (A, B1)

Λ δ Λ Torn+1(A, B3) Torn (A, B1) (b) The long sequence ···→ Λ →δ Λ → Λ → Λ →···δ Torn+1(A, B3) Torn (A, B1) Torn (A, B2) Torn (A, B3) is exact.

{ Λ − } = The sequence Torn+1(A, ) (n 0, 1,...) is universal. Analogous properties Λ − hold for Torn ( ,B).

1.10 Homology and Cohomology of Proﬁnite Groups

ˆ Let R be a commutative proﬁnite ring; usually one is interested in R being Z or one = = of its quotient rings: Zp, ZCˆ, Zπˆ p∈π Zp, Z/nZ, Z/pZ Fp,..., where p is a prime number, π is a set of prime numbers, C is a pseudovariety of ﬁnite groups and n is a natural number.

Cohomology of Proﬁnite Groups

Let G be a profinite group. Consider R as a profinite G-module with trivial action: gr = r, for all g ∈ G, r ∈ R. Then R becomes a profinite [[ RG]] -module. Given a discrete [[ RG]] -module A and a natural number n, define the n-th cohomology group H n(G, A) of G with coefficients in A as the discrete R-module n = n H (G, A) Ext[[ RG]] (R, A). Define AG ={a | a ∈ A,ga = a, ∀g ∈ G}. Then AG is an [[ RG]] -submodule of A. We call AG the submodule of fixed points of A. This submodule coincides with Hom[[ RG]] (R, A). It is convenient to restate explicitly the properties mentioned in Sect. 1.9 for { n − } = { n − } = Ext[[ RG]] (R, ) (n 0, 1,...), using the notation H (G, ) (n 0, 1,...). It is a universal sequence of functors and has the following properties.

0 G 1.10.1 H (G, A) = Hom[[ RG]] (R, A) = A . 1.10 Homology and Cohomology of Proﬁnite Groups 21

1.10.2 Each H n(G, −) is an additive functor.

1.10.3 H n(G, Q) = 0 for every injective discrete [[ RG]] -module Q and n ≥ 1.

1.10.4 For each short exact sequence 0 −→ A1 −→ A2 −→ A3 −→ 0 in DMod([[ RG]] ), there exist connecting homomorphisms n n+1 δ : H (G, A3) −→ H (G, A1) for all n ≥ 0, such that the sequence

0 0 0 δ 1 1 0 → H (G, A1) → H (G, A2) → H (G, A3) → H (G, A1) → H (G, A2) →··· is exact; and For every commutative diagram

0 A1 A2 A3 0

α β γ

0 A1 A2 A3 0 in DMod([[ RG]] ) with exact rows, the following diagram commutes for every n ≥ 0

n δ n+1 H (G, A3) H (G, A1)

H n(G,γ ) H n+1(G,α)

n δ n+1 H (G, A3) H (G, A1).

n = n As mentioned above, to compute H (G, A) Ext[[ RG]] (R, A) explicitly one can start, for example, by ﬁnding a projective resolution for R, applying to it the functor Hom[[ RG]] (−,A) and then computing the n-th cohomology group of the resulting sequence. In this case, there are standard ways of doing all of this and they are not hard. Rather than specifying all the steps, we just describe the end result; this will permit us to recall easily the deﬁnitions of certain special maps that are needed later in the book. For a natural number n,letCn(G, A) consist of all continuous maps

n+1 f : Gn+1 = G × ··· ×G −→ A such that

f(xx0,xx1,...,xxn) = xf (x0,x1,...,xn) ∀x,xi ∈ G. Then Cn(G, A) is a discrete R-module. Consider the sequence of R-modules and R-homomorphisms

+ ∂n 1 + 0 −→ C0(G, A) −→···−→Cn(G, A) −→ Cn 1(G, A) −→··· , 22 1 Preliminaries where n+1 n+1 i ∂ f (x0,x1,...,xn+1) = (−1) f(x0,...,xˆi,...xn+1) i=0

(the symbol xˆi indicates that xi is to be omitted). Then one has the following explicit description:

n = n = n+1 n H (G, A) Ext[[ RG]] (R, A) Ker ∂ /Im ∂ . (1.5) The elements in Ker(∂n+1) are called n-cocycles, and the elements of Im(∂n), n- coboundaries. Using (1.5) one obtains a useful description of, for example, the ﬁrst cohomology group H 1(G, A). To explain this, recall that a derivation d : G −→ A from a proﬁnite group G to G-module A is a continuous function such that d(xy) = xd(y) + d(x), for all x,y ∈ G. We denote the abelian group of derivations from G to A by Der(G, A).Ifa ∈ A,the map

da : G −→ A given by the formula da(x) = xa−a (x ∈ G) is a derivation, called the inner derivation determined by a. The inner derivations form a subgroup of Der(G, A) which is denoted by Ider(G, A). Then one has

1.10.5 With the notation above, H 1(G, A) = Der(G, A)/Ider(G, A).

Derivations are useful in several other contexts in this book. To explain this recall ﬁrst that the augmentation ideal (( I G )) of the complete group ring [[ RG]] is the kernel of the continuous ring homomorphism (the augmentation map) ε :[[RG]] −→ R that sends R identically to R, and ε(g) = 1, for every g ∈ G.

1.10.6 (a) (( I G )) is a proﬁnite free R-module on the pointed topological space G − 1 ={x − 1 | x ∈ G}, where 0 is the distinguished point of G − 1. (b) If T is a proﬁnite subspace generating G such that 1 ∈ T , then (( I G )) is generated by the pointed space T − 1 ={t − 1 | t ∈ T }, as an [[ RG]] -module.

Next we see how derivations give rise to homomorphisms. 1.10 Homology and Cohomology of Proﬁnite Groups 23

1.10.7 There is a natural isomorphism

ϕ : Der(G, A) −→ Hom[[ RG]] (( I G )) , A deﬁned by (ϕ(d))(x − 1) = d(x) (d ∈ Der(G, A), x ∈ G).

Special Maps in Cohomology

Let H be a closed subgroup of a proﬁnite group G.Adiscrete[[ RG]] -module A is also a discrete [[ RH]] -module. For each natural number n, one has a homomorphism = G : n −→ n Res ResH H (G, A) H (H, A) called restriction. One can deﬁne these mappings using the universality of the sequence of functors H n(H, −) by specifying Res in dimension zero: it is just the inclusion AG → AH . Or, more explicitly, one can describe Res for each dimension if we use the formula (1.5): Let σ : Gn+1 −→ A be an n-cocycle represent- inganelementσ¯ ∈ H n(G, A); then a representative n-cocycle ρ : H n+1 −→ A of Res(σ)¯ ∈ H n(H, A) is given by

ρ(x0,...,xn) = σ(x0,...,xn), (x0,...,xn ∈ H).

Assume now that K is a closed normal subgroup of a proﬁnite group G, and let A be a discrete [[ RG]] -module. Then AK becomes a discrete [[ R(G/K)]] -module in a natural way:

(xK)(a) = xa, x ∈ G, a ∈ AK .

Deﬁne a homomorphism

= G/K : n K −→ n Inf InfG H G/K, A H (G, A), called inﬂation, as follows. In dimension n = 0, deﬁne

Inf : H 0 G/K, AK = AK G/K −→ H 0(G, A) = AG to be the identity mapping. Assume n>0, and let σ ∈ Cn(G/K, AK ) represent an element σ¯ of H n(G/K, AK ), i.e., σ : (G/K)n+1 −→ AK is an n-cocycle. Then Inf(σ)¯ is represented by the n-cocycle

ρ : Gn+1 −→ A given by

ρ(x0,...,xn) = σ(x0K,...,xnK). 24 1 Preliminaries

Homology of Proﬁnite Groups

Let G be a profinite group and let B be a profinite right [[ RG]] -module. Define the n-th homology group Hn(G, B) of G with coefficients in B by the formula = [[ RG]] Hn(G, B) Torn (B, R). In dimension zero one has

1.10.8 ∼ def H0(G, B) = B/B((IG)) = B/ bg − b | b ∈ B,g ∈ G = BG.

Duality Homology-Cohomology

Given a Λ-module M (discrete or profinite), consider the abelian group (the Pon- tryagin dual of M) ∗ M = Hom(M, Q/Z) of all continuous homomorphism from M to Q/Z (as abelian groups) with the compact-open topology. M∗ is profinite if M is discrete, and it is discrete if M is profinite. Define a right action of Λ on M∗ by

f λ (m) = f(λm) f ∈ M∗,λ∈ Λ,m ∈ M .

Then M∗ becomes a right Λ-module. Also, if M is a projective proﬁnite Λ-module, then M∗ becomes an injective discrete Λ-module, and vice versa.

1.10.9 Let G be a proﬁnite group and let B be a right [[ Zˆ G]] -module. Then

n ∗ Hn(G, B) and H G, B (n ∈ N) are Pontryagin dual, where B∗ denotes the Pontryagin dual of B.

This duality permits the automatic translation of information in cohomology to homology, or vice versa. For example, one can easily state the dual results of 1.10.1Ð 1.10.4 for homology groups. The dual of the homomorphism Res described above is called corestriction: = H : −→ Cor CorG Hn(H, B) Hn(G, B). Similarly, the dual of Inf is called coinﬂation, and denoted Coinf. In dimension 1, we have the following useful descriptions when considering convenient coefﬁcient modules: 1.11 (Co)homological Dimension 25

1.10.10 (a) Let G be a proﬁnite group and let G act on Zˆ trivially. Then there is a natural isomorphism ˆ ∼ H1(G, Z) = G/[G, G].

(b) Let G be a pro-p group and let G act on Fp trivially. Then there is a natural isomorphism ∼ H1(G, Fp) = G/Φ(G), where Φ(G) = Gp[G, G] is the Frattini subgroup of G.

(Co)induced Modules and Shapiro’s Lemma

Let H ≤ G be profinite groups and R a commutative profinite ring. Let B be a profinite right [[ RH]] -module. Then G = ⊗ [[ ]] IndH (B) B [[ RH]] RG has a natural structure as a right profinite [[ RG]] -module and it is called an induced module.Dually,ifA is a left discrete [[ RH]] -module, one defines the coinduced module

G = [[ ]] CoindH (A) Hom[[ RH]] RG ,A . It has a natural structure as a left discrete [[ RG]] -module. The following result is traditionally known as ‘Shapiro’s Lemma’.

1.10.11

(a)

G =∼ ≥ Hn G, IndH (B) Hn(H, B) (n 0). (b) Dually,

n G =∼ n ≥ H G, CoindH (A) H (H, A) (n 0).

1.11 (Co)homological Dimension

Let G be a proﬁnite group and let p be a prime number. If A is an abelian group, then Ap denotes its p-primary component (the subgroup of those elements of A of n order p ,forsomen). If A = Ap, we say that A is p-primary. The cohomological p-dimension

cdp(G) 26 1 Preliminaries

k of G is the smallest non-negative integer n such that H (G, A)p = 0 for all k>n ˆ and A ∈ DMod([[ ZG]] ), if such an n exists; otherwise we say that cdp(G) =∞. Dually, one can deﬁne the homological dimension of G, denoted hdp(G).Wehave cdp(G) = hdp(G).

1.11.1 If Gp is a p-Sylow subgroup of a proﬁnite group G, then cdp(G) = cdp(Gp).

1.11.2 Let G be a proﬁnite group and let n be a ﬁxed natural number. Then cdp(G) ≤ n if and only if there exists a projective resolution

0 → Pn → Pn−1 →···→P0 → Fp → 0 of Fp in PMod([[ FpG]] ) of length n.

It is often important to compare cohomological dimensions of groups and subgroups. The following result is particularly useful.

1.11.3 Let H ≤c G be proﬁnite groups. Then cdp(H ) ≤ cdp(G). If cdp(G) is ﬁnite and H is open in G, then cdp(H ) = cdp(G).

In the case of pro-p groups, freeness can be characterized in terms of cohomological dimension (as is the case for abstract groups):

1.11.4 Let G beapro-p group. Then G is free pro-p if and only if cdp(G) ≤ 1.

1.11.5 Let G be a proﬁnite group. Then G is projective if and only if cdp(G) ≤ 1 for every prime number p. Part I Basic Theory

In this first part of the book (the next five chapters) we develop the fundamentals of profinite graphs and we relate them to profinite groups. We view a graph as a collection Γ = V ∪ E of vertices V and edges E together with functions d0,d1 defined on Γ that assign to each m ∈ Γ vertices, its ‘origin’ d0(m) and ‘terminus’ d1(m) (a vertex is assumed to be its own origin and terminus). We call such a graph ‘profinite’ if Γ has in addition the structure of a profinite space (i.e., a compact, Hausdorff and totally-disconnected topological space) in such a way that V is a closed subspace of Γ and d0 and d1 are continuous functions. Profinite graphs arise naturally in connection with profinite groups and originally they were introduced precisely as tools in the study of those groups. Here we develop an analogue to the BassÐSerre theory of (abstract) groups and (abstract) graphs. Our development is self-contained and in principle it does not require any knowledge of Bass and Serre’s elegant theory of groups acting on trees (Serre 1980). However, we will frequently mention connections with abstract groups and graphs and the BassÐSerre results; and, in fact, as we will see later in Part III of this book, the interconnections with the BassÐSerre approach will be crucial for applications to abstract groups. In Chap, 2 the concept of a π-tree is introduced, where π is a set of prime numbers; a paradigm of a π-tree is the Cayley graph of a free pro-π group with respect to a basis. Profinite graphs are just inverse limits of finite graphs, but a π-tree need not be the inverse limit of finite trees. In Chaps. 2Ð4 we study profinite groups acting on π-trees, the fundamental group of a profinite graph and simple connectivity of profinite graphs. Unlike the situation for abstract graphs, the class of simply connected profinite graphs does not coincide with the class of π-trees. Free products of profinite groups are studied in Chap. 5. The need to describe the structure of closed subgroups of a free product of even just two profinite groups leads to the concept of a collection of closed subgroups of a profinite group ‘continuously indexed by a profinite space’, and to free products of such collections. More general ‘free constructions’ of profinite groups (e.g., amalgamated products, HNN extensions or, more generally, fundamental groups of graphs of profinite 28 groups) and standard profinite graphs associated with these constructions are considered in Chap. 6. A free profinite product of profinite groups always contains the free factors. But, unlike the abstract case, a profinite fundamental group need not contain the vertex groups of the corresponding graph of profinite groups; when they do, we call the fundamental groups ‘proper’. This difficulty may already be encountered in the case of an amalgamated product of two groups amalgamating a common subgroup. As wewillseeinPartIII of this book, in certain applications to abstract groups, one of the key points is to prove that appropriate profinite fundamental groups are proper. Chapter 2 Profinite Graphs

Unless otherwise specified, in this chapter C is a pseudovariety of finite groups, i.e., a nonempty class of finite groups closed under subgroups, quotients and finite direct products.

2.1 First Notions and Examples

A proﬁnite graph is a proﬁnite space Γ with a distinguished nonempty subset V(Γ), the vertex set of the graph Γ , and two continuous maps

d0,d1 : Γ → V(Γ) whose restrictions to V(Γ) are the identity map idV(Γ) (to simplify the notation, we sometimes write dim, rather than di(m) (m ∈ Γ,i = 0, 1)). This implies that the distinguished subset V(Γ)is necessarily closed. The elements of V(Γ)are called the vertices of Γ , the elements of E(Γ ) = Γ − V(Γ)are the edges of Γ , and d0(e) and d1(e) are the initial and terminal vertices of an edge e, respectively (also called the origin and terminus of e). An edge e with d0(e) = d1(e) = v is called a loop or a loop based at v. We refer to d0 and d1 as the incidence maps of the graph Γ . Observe that a profinite graph is also a graph in the usual sense, or, more precisely, an oriented graph (see Appendix A), if we dispense with the topology. The set of edges E(Γ ) of a profinite graph Γ need not be a closed subset of Γ .IfE(Γ ) is closed (and therefore compact), it is enough to check the continuity of d0 and d1 on V(Γ)and E(Γ ) separately, since then V(Γ)and E(Γ ) are disjoint and clopen. Associated with each edge e of Γ we introduce symbols e1 and e−1. We identify 1 −1 e with e. Define incidence maps for these symbols as follows: d0(e ) = d1(e) −1 and d1(e ) = d0(e). Given vertices v and w of Γ ,apath pvw from v to w is a ε1 εm ≥ ∈ =± = finite sequence e1 ,...,em , where m 0, ei E(Γ ), εi 1 (i 1,...,m)such ε1 = εm = εi = εi+1 = − that d0(e1 ) v, d1(em ) w and d1(ei ) d0(ei+1 ) for i 1,...,m 1. Such a path is said to have length m. Observe that a path is always meant to be finite. The underlying graph of the path pvw consists of the edges e1,...,em and their

© Springer International Publishing AG 2017 29 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_2 30 2 Proﬁnite Graphs vertices di(ej ) (i = 0, 1; j = 1,...,m). The path pvw is called reduced if whenever ei = ei+1, then εi = εi+1, for all i = 1,...,m− 1.

Example 2.1.1 (a) A finite abstract graph Γ (see Appendix A) with the discrete topology is a profinite graph. (b) Let N ={0, 1, 2,...} and N˜ ={ñ | n ∈ N} be copies of the set of natural numbers (with the discrete topology). Define I = N ∪. N˜ ∪. {∞} to be the one-point compactification of the space N ∪. N˜ . Recall that then in the topology of I each set {n} and {ñ} is open (n ∈ N), and the basic open neighbourhoods of ∞ are the complements of finite subsets of N ∪. N˜ . Clearly I is a profinite space. We make I into a profinite graph by setting V(I)= N ∪. {∞}, E(I) = N˜ , d0(n)˜ = n, d1(n)˜ = n + 1, for n˜ ∈ E(I), and di(n) = n,forn ∈ V(I)(i= 1, 2). 0 123 ∞ ••••··· • 0˜ 1˜ 2˜ Observe that in this case the subset of edges E(I) is open, but not closed in I . (c) Let p be a prime number and let Zp be the additive group of the ring of p-adic integers. Define a graph

Γ = Γ Zp, {1} with set of vertices V = V(Γ)= Zp and whose set of edges is E = E(Γ ) ={(α, 1) | α ∈ Zp}. Then V(Γ)and E(Γ ) are profinite spaces. We define the topology of Γ = V(Γ)∪. E(Γ ) to be the disjoint topology: a subset A of Γ is open if and only if A ∩ V is open in V and A ∩ E is open in E. One easily sees that Γ is a profinite space. Observe that the subset of edges E = E(Γ ) of Γ is both open and closed (clopen) in the topology of Γ . The incidence maps are the continuous maps

di : Γ −→ V(i= 0, 1) defined as d0(α) = α, d0(α, 1) = α and d1(α) = α, d1(α, 1) = α + 1 (α ∈ Zp). With these definitions Γ becomes a profinite graph. [This is an instance of profinite graphs obtained from profinite groups in a standard manner, the so-called Cayley graphs: see Example 2.1.12.] The subgroup of integers Z = 1 is dense in Zp and the topology of Z induced by the topology of Zp is the discrete topology. Let Γ Z, {1} = α ∈ V(Γ) α ∈ Z ∪. (α, 1) α ∈ Z . Then Γ(Z, {1}) is an abstract discrete graph −2 −1 0 12 ··· •••••··· (−2, 1)(−1, 1)(0, 1)(1, 1) which is dense in the profinite graph Γ = Γ(Zp, {1}). 2.1 First Notions and Examples 31

More generally, let β be a fixed element of Zp, and define Γ Z + β,{1} = α ∈ V(Γ) α ∈ Z + β ∪. (α, 1) ∈ E(Γ ) α ∈ Z + β . Then Γ(Z + β,{1}) is an abstract discrete graph β − 2 β − 1 β β + 1 β + 2 ··· •••••··· (β − 2, 1)(β− 1, 1)(β,1)(β+ 1, 1) which is also dense in the profinite graph Γ = Γ(Zp, {1}). Note that Γ(Zp, {1}) is a disjoint union of uncountably many abstract discrete graphs of the form Γ(Z + β,{1}): . Γ Zp, {1} = Γ Z + βλ, {1} , λ∈Λ where {βλ | λ ∈ Λ} is a complete set of representatives of the cosets of the subgroup Z in the group Zp.

Let Γ and be profinite graphs. A qmorphism or a quasi-morphism of profinite graphs or a map of graphs α : Γ → is a continuous map such that dj (α(m)) = α(dj (m)), for all m ∈ Γ and j = 0, 1. If in addition α(e) ∈ E() for every e ∈ E(Γ ), we say that α is a morphism. The composition of qmorphisms of profinite graphs is again a qmorphism, so that profinite graphs and their qmorphisms form a category. Similarly profinite graphs and their morphisms form a category. If α is a surjective (respectively, injective, bijective) qmorphism, we say that α is an epimorphism (respectively, monomorphism, isomorphism). An isomorphism α : Γ → Γ of the graph Γ to itself is called an automorphism. Note that a monomorphism of graphs sends edges to edges, and hence it is always a morphism. A nonempty closed subset Γ of a profinite graph is called a profinite subgraph of if whenever m ∈ Γ , then dj (m) ∈ Γ(j= 0, 1). The equality dj (α(m)) = α(dj (m)) (j = 0, 1; m ∈ Γ)implies that a qmorphism of profinite graphs maps vertices to vertices. However, the next example shows that a qmorphism can map an edge to a vertex.

Example 2.1.2 (Subgraph collapsing) Let be a profinite subgraph of a profinite graph Γ . Consider the natural continuous map α : Γ → Γ/ to the quotient space Γ/ with the quotient topology [the points of Γ/ are the equivalence classes of the relation ∼ on Γ defined as follows: if m, m ∈ Γ , then m ∼ m if and only if either m = m or m, m ∈ ;ifm ∈ Γ , then α(m) is the equivalence class of m; a subset U of Γ/ is open if α−1(U) is open in Γ ]. Define a structure of profinite graph on the space Γ/ as follows: V(Γ/)= α(V (Γ )), d0(α(m)) = α(d0(m)), d1(α(m)) = α(d1(m)), for all m ∈ Γ . Then clearly α is a qmorphism of graphs and Γ/ becomes a quotient graph of Γ . We shall say that Γ/is obtained from Γ by collapsing to a point. Observe that α maps any edge of Γ which is in toavertexofΓ/. 32 2 Profinite Graphs

We note that if α : Γ → is an epimorphism of profinite graphs, then has the quotient topology (i.e., for A ⊆ , one has that A is open in if and only if α−1(A) is open in Γ ), since Γ and are compact Hausdorff spaces. We then say that is a quotient graph of Γ and α is a quotient qmorphism of graphs. If Γ is a profinite graph and ϕ : Γ → Y is a continuous surjection onto a profinite space Y , there is no assurance that there exists a profinite graph structure on Y so that ϕ is a qmorphism of graphs. The following construction provides necessary and sufficient conditions for this to happen.

Construction 2.1.3 Let Γ be a proﬁnite graph and let ϕ : Γ → Y be a continuous surjection onto a proﬁnite space Y . Then we construct a quotient qmorphism of graphs

ϕ˜ : Γ → Γϕ with the following properties.

(a) There is a continuous surjection of topological spaces ψϕ : Γϕ → Y such that the diagram ϕ Γ Y

ϕ˜ ψϕ Γϕ commutes. (b) If Y admits a profinite graph structure so that ϕ is a qmorphism, then ψϕ is an isomorphism of profinite graphs. (c) Consequently, there exists a profinite graph structure on Y such that ϕ is a qmorphism of graphs if and only if whenever m, m ∈ Γ with ϕ(m) = ϕ(m ),

then ϕd0(m) = ϕd0(m ) and ϕd1(m) = ϕd1(m ). If this is the case, then that structure is unique (isomorphic to Γϕ) and the incidence maps of Y are deﬁned by diϕ(m) = ϕdi(m) (m ∈ Γ,i = 0, 1). (d) If E(Γ ) is a closed subset of Γ and ϕ(E(Γ )) ∩ ϕ(V (Γ )) =∅, then ϕ˜ is a morphism of proﬁnite graphs and ψϕ(E(Γϕ)) ∩ ψϕ(V (Γϕ)) =∅.

To construct Γϕ, deﬁne a map ϕ˜ : Γ −→ Y × Y × Y by

ϕ(m)˜ = ϕ(m),ϕd0(m), ϕd1(m) (m ∈ Γ).

Let Γϕ =˜ϕ(Γ ). Then Γϕ admits a unique graph structure such that ϕ˜ : Γ → Γϕ is a ˜ ˜ qmorphism of graphs, namely one is forced to deﬁne the incidence maps d0 and d1 of Γϕ by

˜ d0 ϕ(m),ϕd0(m), ϕd1(m) = ϕd0(m), ϕd0(m), ϕd0(m) (m ∈ Γ) 2.1 First Notions and Examples 33 and

˜ d1 ϕ(m),ϕd0(m), ϕd1(m) = ϕd1(m), ϕd1(m), ϕd1(m) (m ∈ Γ) (one easily checks that these are well defined, and that ϕ˜ is indeed a qmorphism of profinite graphs). Next note that there exists a unique map ψϕ : Γϕ → Y such that ψϕϕ˜ = ϕ, namely, ψϕ(ϕ(m), ϕd0(m), ϕd1(m)) = ϕ(m). If Y is a profinite graph and ϕ is a qmorphism of profinite graphs, then ψϕ is an isomorphism of graphs because in this case the map ρ : Y → Γϕ given by ρϕ(m) = (ϕ(m), ϕd0(m), ϕd1(m)) is a well-defined qmorphism of graphs and it is inverse to ψϕ. This proves properties (a) and (b). Property (c) is clear. Property (d) is easily verified.

Before stating the following proposition we recall briefly the concept of an inverse limit in the category of graphs (see Sect. 1.1). Let (I, ) be a directed partially ordered set (a directed poset). An inverse system of profinite graphs {Γi,ϕij ,I} over the directed poset I consists of a collection of profinite graphs Γi indexed by I and qmorphisms of profinite graphs ϕij : Γi → Γj , whenever i j, in such a way that ϕii = Idi , for all i ∈ I , and ϕjkϕij = ϕik, whenever i j k. The inverse limit (or projective limit) of such a system = Γ lim←− Γi i∈I = is the subset of i∈I Γi consisting of those tuples (mi) with ϕij (mi) mj , whenever i j. Such an inverse limit is in a natural way a profinite graph whose space of vertices is = V(Γ) lim←− V(Γi). i∈I

Observe that the natural projections ϕi : Γ → Γi are qmorphisms of profinite graphs. Note that if each ϕij is a morphism, then so are the canonical projections ϕi . Let Γ be a profinite graph and consider the set R of all open equivalence relations R on the set Γ (i.e., the equivalence classes xR are open for all x ∈ Γ ). For R ∈ R, denote by ϕR : Γ → Γ/R the corresponding quotient map as topological spaces. One defines a partial ordering on R as follows: for R1,R2 ∈ R, we say that : → R1 R2 if there exists a map ϕR1,R2 Γ/R1 Γ/R2 such that the diagram

Γ/R1 ϕR1

Γ ϕR1,R2

ϕR2 Γ/R2 commutes. Then (cf. RZ, Theorem 1.1.2) (R, ) is in fact a directed poset, { } R Γ/R,ϕR1,R2 is an inverse system over , and, as topological spaces, the collection 34 2 Profinite Graphs of quotient maps {ϕ | R ∈ R} induces a homeomorphism from Γ to lim ∈RΓ/R; R ←− R in fact we identify these two spaces by means of this homeomorphism and write = Γ lim←− Γ/R. (2.1) R∈R Consider now the subset R of R consisting of those R ∈ R such that Γ/R admits a graph structure (which is unique according to part (c) of Construction 2.1.3)so that ϕR : Γ → Γ/R is a qmorphism of profinite graphs. We check next that the poset (R , ) is directed. Indeed, let R1,R2 ∈ R . Since R is directed, there exists an R ∈ R such that R R1,R2.LetϕR : Γ → Γ/R be the corresponding quotient : → = ˜ map. Let ΓϕR and ϕR Γ ΓϕR be as in Construction 2.1.3. Then ΓϕR Γ/R, ˜ −1 where R is the equivalence relation on Γ whose equivalence classes are {ϕR (x) | ∈ } ˜ ∈ R ˜ ˜ x ΓϕR . Clearly R and R R; hence R R1,R2, as needed. ∈ R : → Observe that if R1,R2 and R1 R2, then the map ϕR1,R2 Γ/R1 Γ/R2 { } is in fact a qmorphism of finite graphs. Therefore the collection Γ/R,ϕR1,R2 of all finite quotient graphs of Γ is an inverse system of finite graphs and qmorphisms over the directed poset R .

Proposition 2.1.4 Let Γ be a proﬁnite graph. (a) Γ is the inverse limit of all its ﬁnite quotient graphs: = Γ lim←− Γ/R. R∈R Consequently = V(Γ) lim←− V(Γ/R). R∈R (b) If the subset E(Γ ) of edges of Γ is closed, then a directed subposet R of R ∈ R : can be chosen so that whenever R1,R2 with R1 R2, then ϕR1,R2 Γ/R1 → Γ/R2 is a morphism of graphs and = Γ lim←− Γ/R. R∈R Consequently, = = V(Γ) lim←− V(Γ/R) and E(Γ ) lim←− E(Γ/R). R∈R R∈R

Proof (a) In view of (2.1) one simply has to show that R is cofinal in R, i.e., one has to show that whenever R ∈ R, there exists an R ∈ R with R R. But this is clear from property (a) of Construction 2.1.3. (b) Suppose that E(Γ ) is closed. Then Γ = V(Γ)∪. E(Γ ) and V(Γ)and E(Γ ) are clopen subsets of Γ .LetR˜ be the subset of R consisting of those equivalence relations R ∈ R whose equivalence classes xR are contained in either E(Γ ) 2.1 First Notions and Examples 35 or V(Γ); this implies that if ϕR : Γ → Γ/R is the canonical projection, then ˜ ϕR(V (Γ )) ∩ ϕR(E(Γ )) =∅. Then one shows that R is cofinal in R, so that = Γ lim←− Γ/R. R∈R˜ One can argue now as in part (a); we just indicate the main points: let R be the subset of R˜ consisting of those equivalence relations R such that Γ/R has the structure of a graph in such a way that ϕR : Γ → Γ/R is a morphism of profinite graphs; note that R is also a subset of R ; using property (d) of Construction 2.1.3 one shows that R is cofinal in R˜ , and hence the result easily follows as above.

Lemma 2.1.5 Let {Γi,ϕij ,I} be an inverse system of proﬁnite graphs and qmorphisms over a directed poset I , and set = Γ lim←− Γi. (2.2) i∈I Let ρ : Γ → be a qmorphism into a ﬁnite graph . Then there exists a k ∈ I such that ρ factors through Γk, i.e., there exists a qmorphism ρ : Γk → such that

ρ = ρ ϕk, where ϕk : Γ → Γk is the projection.

Proof For i ∈ I denote by Ri the set of all equivalence relations R of Γi such that the quotient Γi/R is a finite discrete graph and the natural projection Γi → Γi/R is a qmorphism. Define an ordering on the set of pairs A = (i, R) i ∈ I,R ∈ Ri by setting (i, Ri) (j, Rj ),ifi j and (ϕij × ϕij )(Ri) ⊆ Rj . Let us prove that (A, ) is a directed poset. Fix i, j ∈ I and Ri ∈ Ri , Rj ∈ Rj . Since I is a directed poset, there exists some k ∈ I with k i, j. By Proposition 2.1.4, Γk is the inverse limit of all its finite quotient graphs; therefore there exists an Rk ∈ Rk with (ϕki × ϕki)(Rk) ⊆ Ri and (ϕkj × ϕkj )(Rk) ⊆ Rj , so that (k, Rk) (i, Ri), (j, Rj ), as needed. Now it is easy to see that = Γ lim←− Γi/R. (i,R)∈A

Thus from now on we may assume that each Γi in the decomposition (2.2)isfinite. Assume first that each projection ϕi : Γ → Γi is surjective. Let S be the equivalence relation on Γ whose equivalence classes are the clopen sets ρ−1(m), m ∈ ; then Γ/S = and ρ is the natural projection Γ → Γ/S. Similarly, for i ∈ I , let Si be the equivalence relation on Γ whose equivalence classes are the clopen −1 ∈ → = sets ϕi (m), m Γi , so that ϕi is the natural projection Γ Γ/Si . Since Γ lim ∈ Γ , we have that S is the trivial equivalence relation, i.e., S = D, ←− i I i i∈I i i∈I i where D is the diagonal subset of Γ × Γ . Note that S and Si (i ∈ I)are clopen subsets of Γ × Γ . Hence, it follows from the compactness of Γ × Γ that there exists ⊆ a finite subset F of I such that j∈F Sj S. Since the poset I is directed, there 36 2 Profinite Graphs ∈ ⊆ ⊆ exists a k I with Sk j∈F Sj S. This means that there exists a qmorphism of graphs ρk : Γk = Γ/Sk → = Γ/S such that ρ = ρkϕk.

Consider now a general ϕi . By the above, there exists some k ∈ I andaqmor- phism of graphs ρk : ϕk (Γ ) → such that ρ = ρk ϕk . Since Γk is ﬁnite, there exists a k k such that ϕkk (Γk) ⊆ ϕk (Γ ). Then ρ = ρk ϕkk is the required qmorphism.

An alternative proof of Lemma 2.1.5 above can be obtained along the lines of the proof of Lemma 1.1.16 in RZ. A profinite graph Γ is said to be connected if whenever ϕ : Γ → A is a qmorphism of profinite graphs onto a finite graph, then A is connected as an abstract graph (see Sect. A.1 in Appendix A).

Proposition 2.1.6 (a) Every quotient graph of a connected proﬁnite graph is connected. (b) If = Γ lim←− Γi i∈I

and each Γi is a connected profinite graph, then Γ is a connected profinite graph. (c) Let Γ be a connected profinite graph. If |Γ | > 1, then Γ has at least one edge. Furthermore, if the set of edges E(Γ ) of Γ is closed in Γ , then for any vertex v ∈ V(Γ), there exists an edge e ∈ E(Γ ) such that either v = d0(e) or v = d1(e). (d) Let Γ be a profinite graph, and let be a connected profinite subgraph of Γ . Consider the quotient graph Γ/ obtained by collapsing to a point and let α : Γ → Γ/ be the natural projection. Then the inverse image Λ˜ = α−1(Λ) in Γ of a connected profinite subgraph Λ of Γ/is a connected profinite subgraph.

Proof Part (a) is obvious. Let A be a finite quotient graph of Γ . Then (see Lemma 2.1.5) there exists an i ∈ I such that A is also a quotient graph of Γi .It follows that A is connected, proving (b). To check the first assertion in (c) observe that by Proposition 2.1.4, Γ has a finite quotient graph with at least two elements; since such a finite quotient graph is connected, it has at least one edge, and hence so does Γ . To check the second assertion in (c), write Γ as an inverse limit Γ = lim ∈ Γ of finite quotient graphs ←− i I i Γi in such a way that = E(Γ ) lim←− E(Γi) i∈I

(see Proposition 2.1.4(b)). For i ∈ I ,letϕi : Γ → Γi denote the canonical projection, and if i, j ∈ I with i j,letϕij : Γi → Γj denote the canonical morphism. Put vi = ϕi(v) (i ∈ I). Since Γi is a connected ﬁnite graph, the set Si = 2.1 First Notions and Examples 37

−1 ∪ −1 d0 (vi) d1 (vi) of edges of Γi starting or ending at vi is nonempty; moreover, ϕij (Si) ⊆ Sj . Hence the collection {Si}i∈I is an inverse system of nonempty finite sets. Thus = ∅ lim←− Si i∈I (see Sect. 1.1). Let e ∈ lim ∈ S . Then e is an edge of Γ with either d (e) = v or ←− i I i 0 d1(e) = v. (d) This is clear if Γ is finite. Write = Γ lim←− Γi, i∈I where each Γi is a connected finite quotient graph of Γ (see Proposition 2.1.4(a)). Let i be the image of in Γi under the canonical projection. Then = = lim←− i and Γ/ lim←− Γi/i. i∈I i∈I ˜ Let Λi be the image of Λ in Γi/i , and denote by Λi its inverse image in Γi . Since ˜ ˜ ˜ Λ = lim ∈ Λ , Λ is connected according to part (b). ←− i I i

Lemma 2.1.7 (a) Let D be an abstract subgraph of a profinite graph Γ . Then the topological closure D¯ of D in Γ is a profinite graph. If D is connected as an abstract graph (see Sect. A.1 in Appendix A), then D¯ is a connected profinite graph. { | ∈ } (b) Let j j J be a collection of connected profinite subgraphs of a profinite = ∅ = graph Γ . If j∈J j , then j∈J j is connected. ¯ Proof To prove (a), let m ∈ D. By the continuity of di , di(m) ∈ V(D)(i= 1, 2), so that D¯ is a (profinite) graph with V(D)¯ = V(D).Ifϕ : D¯ → A is a qmorphism of profinite graphs onto a finite graph, then ϕ(D)¯ = ϕ(D) = A by continuity. Since D is a connected abstract graph, one easily checks that ϕ(D) is a finite connected graph; hence D¯ is a connected profinite graph. This proves (a). For part (b) note that if α : → A is a qmorphism onto a finite graph A, then ∈ = α(j ) is a connected finite subgraph of A(j J). Since A j∈J α(j ), and = ∅ j∈J α(j ) , it follows that A is a connected abstract graph.

Example 2.1.8 (A connected profinite graph which is not connected as an abstract graph and with a vertex with no edge beginning or ending at it) Let I be the graph considered in Example 2.1.1(b): I = N ∪. N˜ ∪. {∞} is the one-point compactification of a disjoint union of two copies N and N˜ ={ñ | n ∈ N} of the natural numbers; ˜ V(I)= N ∪. {∞}, E(I) = N, d0(n)˜ = n, d1(n)˜ = n + 1forn˜ ∈ E(I), and di(n) = n for n ∈ V(I)(i= 1, 2). 0 123 ∞ ••••··· • 0˜ 1˜ 2˜ 38 2 Profinite Graphs

Then I is a connected profinite graph; to see this consider the connected finite graphs In 0 1 2 n − 1 n • • • ··· • • 0˜ 1˜ n− 1 ˜ ˜ with vertices V(In) ={0, 1, 2,...,n} and edges E(In) ={0, 1,...,n − 1} such that ˜ ˜ d0(i) = i, d1(i) = i + 1(i = 0,...,n− 1) and dj (i) = i (i = 0,...,n; j = 0, 1). If n ≤ m, define ϕm,n : Im → In to be the map of graphs that sends the segment [0,n] identically to [0,n], and the segment [n, m] to the vertex n. Then (In,ϕm,n) is an inverse system of graphs, and = I lim←− In, n∈N where ∞=(n)n∈N. Hence I is a connected profinite graph. We observe that there is no edge e of I which has ∞ as one of its vertices; and so I is not connected as an abstract graph.

Lemma 2.1.9 Let Γ = Γ1 ∪. Γ2 be a profinite graph which is the disjoint union of two open profinite subgraphs Γ1 and Γ2; then Γ is not connected. In particular, a profinite graph that contains two different vertices and no edges is not connected.

Proof Collapse Γ1 to a point v1 and Γ2 to a different point v2 (see Example 2.1.2), ˜ to get a disconnected ﬁnite quotient graph Γ ={v1}∪. {v2} consisting of two vertices and no edges.

A maximal connected profinite subgraph of a profinite graph Γ is called a connected profinite component of Γ .

Proposition 2.1.10 Let Γ be a profinite graph. (a) Let m ∈ Γ . Then there exists a unique connected profinite component of Γ containing m, which we shall denote by Γ ∗(m). (b) Any two connected profinite components of Γ are either equal or disjoint. (c) Γ is the union of its connected profinite components.

Proof Part (c) follows from (a). Part (b) follows from (a) and Lemma 2.1.7(b). To prove (a) observe first that the result is obvious if Γ is finite. By Proposi- tion 2.1.4, Γ can be represented as an inverse limit lim ∈ Γ of finite quotient ←− i I i graphs. For i ∈ I ,letϕi : Γ → Γi denote the projection. Since the image of a ∗ connected profinite graph is connected, the graphs Γ (ϕi(m)) form an inverse sys- i ∗ tem. It suffices to show that the profinite subgraph lim ∈ Γ (ϕ (m)) of Γ is the ←− i I i i connected profinite component of Γ containing m. This profinite subgraph is connected by Proposition 2.1.6(b). If Γ is a connected profinite subgraph of Γ contain- ∗ ing m, then Γ = lim ∈ ϕ (Γ ). Therefore ϕ (Γ ) ⊆ Γ (ϕ (m)) for all i ∈ I . Hence ←− i I i i i i 2.1 First Notions and Examples 39

∗ ∗ Γ ⊆ lim ∈ Γ (ϕ (m)); therefore lim ∈ Γ (ϕ (m)) is maximal connected contain- ←− i I i i ←− i I i i ing m, as desired. The uniqueness of connected proﬁnite components containing m follows from Lemma 2.1.7(b).

Exercise 2.1.11 (a) Let be a profinite graph. Define the space of connected profinite components of as a quotient space /∼, where ∼ is the equivalence relation defined ∗ ∗ as follows: m1 ∼ m2 if and only if (m1) = (m2). Prove that /∼ is a profinite space. [Hint: write as an inverse limit of finite quotient graphs.] (b) Let be a profinite subgraph of a profinite graph Γ . Define the operation of collapsing the connected profinite components of to points as a natural mapping to the quotient space Γ/∼, where ∼ is the equivalence relation defined as ∗ ∗ follows: m1 ∼ m2 if m1 = m2,form1,m2 ∈ Γ − ,or (m1) = (m2) for m1,m2 ∈ . Prove that Γ/∼ is a profinite quotient graph of Γ .

Example 2.1.12 (The Cayley graph) Let G be a profinite group (whose operation is denoted as multiplication and whose identity element is denoted by 1) and let X be a closed subset of G. Put X˜ = X ∪{1}. Define the Cayley graph Γ(G,X)of G with respect to the subset X as follows: Γ(G,X)= G × X,˜ where G × X˜ has the product topology. Define the space of vertices of Γ(G,X) to be V(Γ(G,X))={(g, 1) | g ∈ G}. We identify this space of vertices with G by means of the homeomorphism (g, 1) → g(g∈ G). Finally, the incidence maps

˜ d0,d1 : Γ(G,X)= G × X −→ V Γ(G,X) = G are deﬁned by

d0(g, x) = g and d1(g, x) = gx, g ∈ G, x ∈ X ∪{1} .

Clearly d0 and d1 are continuous and they are the identity map when restricted to V(Γ(G,X))={(g, 1) | g ∈ G}=G. Therefore the Cayley graph Γ(G,X)is a proﬁnite graph. Note that the space of edges is E(Γ (G,X)) = Γ(G,X)− V(Γ(G,X))= G × (X −{1}):

(g,x) g gx, where x ∈ X −{1}. It is a closed (and hence clopen) subset of Γ(G,X) if and only if 1 is an isolated point of X˜ . Observe that if 1 ∈/ X, then V(Γ(G,X))= G and E(Γ (G,X)) = G × X, and in this case E(Γ (G,X)) is clopen. If 1 ∈ X, then X˜ = X. If 1 is in X and it is an isolated point of X (for example, if X is finite), then X −{1} is also a closed subspace and we have Γ(G,X)= Γ(G,X−{1}). Note that the Cayley graph Γ(G,X)does not contain loops since the elements of the form (g, 1) are vertices by definition. 40 2 Profinite Graphs

Let ϕ : G → H be a continuous homomorphism of proﬁnite groups and let X be a closed subset of G. Put Y = ϕ(X). Then ϕ induces a qmorphism of the corresponding Cayley graphs

ϕ˜ : Γ(G,X)−→ Γ(H,Y).

In particular, if U is an open normal subgroup of G and XU = ϕU (X), where ϕU : G → G/U is the canonical epimorphism, then ϕU induces a corresponding epimorphism of Cayley graphs ϕU : Γ(G,X)→ Γ(G/U,XU ). One easily checks that

= Γ(G,X) lim←− Γ(G/U,XU ) UoG is a decomposition of Γ(G,X)as an inverse limit of ﬁnite Cayley graphs.

Example 2.1.13 (An infinite connected profinite graph all of whose proper connected profinite subgraphs are finite) Let Γ = Γ(Z, {1}) be the Cayley graph of the free profinite group Z of rank one with respect the subset {1}. Then

= { } Γ lim←− Γ Z/nZ, 1 , n≥2 with canonical maps

ϕmn : Γ Z/mZ, {1} −→ Γ Z/nZ, {1} (n|m).

Let

ϕn : Γ −→ Γ Z/nZ, {1} denote the projection (n ∈ N). Assume that is a connected proper profinite subgraph of Γ . Put n = ϕn(Γ ). Then n is a connected subgraph of the finite graph Γ(Z/nZ, {1}). = ∈ = { } Since Γ , there exists some n0 N such that n0 Γ(Z/n0Z, 1 ).Ob- serve that for every m ∈ N with n |m, the connected components of ϕ−1 ( ) are 0 mn0 n0 | |=| | isomorphic to n0 . Therefore, m n0 . Thus is finite. It is easy to check that if is a proper connected subgraph of Γ with t + 1 vertices, then there exists a γ ∈ Z such that the vertices of are γ,γ + 1,...,γ + t and with edges (γ, 1), (γ + 1, 1),...,(γ + t − 1, 1):

γ γ + 1 γ + 2 γ + t − 1 γ + t • • • ··· • • (γ, 1)(γ+ 1, 1)(γ+ t − 1, 1) 2.2 Groups Acting on Proﬁnite Graphs 41

2.1.14 Circuits. Let ε = (ε1,...,εn), where εi =±1(i = 1,...,n) and n ≥ 1isa natural number. Deﬁne Circn(ε) to be a graph with n vertices (that we take to be the elements of Z/nZ) and n edges e1,...,en

12e2

e1 e3

Circn(ε) : 0 3 . en . . n − 1 such that d0(ei) = i − 1 and d1(ei) = i,ifεi = 1, and d0(ei) = i and d1(ei) = i − 1, if εi =−1. We refer to a graph of the form Circn(ε) as a circuit of length n or as a n-circuit. A circuit of length 1 is a loop. Note that if n ≥ 2 and ε = (1,...,1), then Circn(ε) = Γ(Z/nZ, {1}).

2.2 Groups Acting on Proﬁnite Graphs

Let G be a profinite group and let Γ be a profinite graph. We say that the profinite group G acts on the profinite graph Γ on the left, or that Γ is a G-graph,if (i) G acts continuously on the topological space Γ on the left, i.e., there is a continuous map G × Γ → Γ , denoted (g, m) −→ gm, g ∈ G, m ∈ Γ , such that (gh)m = g(hm) and 1m = m, for all g,h ∈ G, m ∈ Γ , where 1 is the identity element of G; and (ii) dj (gm) = gdj (m), for all g ∈ G, m ∈ Γ , j = 0, 1.

Observe that if G acts on Γ , then for a fixed g ∈ G,themapρg : Γ → Γ given by m → gm (m ∈ Γ ) is an automorphism of the graph Γ . Hence (cf. RZ, Remark 5.6.1), G acts on a profinite graph Γ if and only if there exists a continuous homomorphism ρ : G −→ Aut(Γ ), where Aut(Γ ) is the group of automorphisms of Γ as a profinite graph, and where the topology on Aut(Γ ) is induced by the compact-open topology. The kernel of the action of G on Γ is the kernel of ρ, i.e., the closed normal subgroup of G consisting of all the elements g ∈ G such that gm = m, for all m ∈ Γ . One defines actions on the right in a similar manner. We shall consider only left actions in this chapter. Let G be a profinite group that acts continuously on two profinite graphs Γ and Γ . A qmorphism of graphs ϕ : Γ −→ Γ 42 2 Profinite Graphs is called a G-map of graphs if ϕ(gm) = gϕ(m), for all m ∈ Γ,g ∈ G. Assume that a profinite group G acts on a profinite graph Γ and let m ∈ Γ . Define

Gm ={g ∈ G | gm = m} to be the stabilizer (or G-stabilizer, if one needs to specify the group G)ofthe element m. It follows from the continuity of the action and the compactness of G that Gm is a closed subgroup of G. Clearly, ≤ ∈ = Gm Gdj (m), for every m Γ,j 0, 1.

If the stabilizer Gm of every element m ∈ Γ is trivial, i.e., Gm = 1, we say that G acts freely on Γ .Ifm ∈ Γ ,theG-orbit of m is the closed subset Gm ={gm | g ∈ G}. If a profinite group G acts on a profinite graph Γ , then G acts on the profinite space V(Γ)of vertices and G acts on E(Γ ). The space G\Γ ={Gm | m ∈ Γ } of G-orbits with the quotient topology is a profinite space which admits a natural profinite graph structure as follows:

V(G\Γ)= G\V(Γ), dj (Gm) = Gdj (m), j = 0, 1. We say that G\Γ is the quotient graph of Γ under the action of G. The corresponding quotient map Γ −→ G\Γ is an epimorphism of proﬁnite graphs given by m → Gm (m ∈ Γ , g ∈ G). We observe that it sends edges to edges (it is a morphism). If N c G, there is an induced action of G/N on N\Γ deﬁned by (gN)(Nm) = N(gm), g ∈ G, m ∈ Γ. The following result is straightforward.

Lemma 2.2.1 Let a profinite group G act on a profinite graph Γ . (a) Let N be a collection of closed normal subgroups of G filtered from below (i.e., the intersection of any two groups in N contains a group in N ) and assume that = G lim←− G/N. N∈N Then the collection of graphs {N\Γ | N ∈ N } is an inverse system in a natural way and = \ Γ lim←− N Γ. N∈N 2.2 Groups Acting on Profinite Graphs 43

(b) Let N c G. For m ∈ Γ , denote by m the image of m in N\Γ . Consider the natural action of G/N on N\Γ deﬁned above. Then (G/N)m is the image of Gm under the natural epimorphism G → G/N. In particular, if Gm ≤ N, for all m ∈ Γ , then G/N acts freely on N\Γ .

Let G be a profinite group. If {Γi,ϕij ,I} is an inverse system of profinite G- graphs and G-maps over the directed poset I , then = Γ lim←− Γi i∈I is in a natural way a profinite G-graph. Next we show that every profinite G-graph admits a decomposition as an inverse limit of finite G-graphs.

Proposition 2.2.2 Let a proﬁnite group G act on a proﬁnite graph Γ . (a) Then there exists a decomposition = Γ lim←− Γi i∈I

of Γ as the inverse limit of a system of ﬁnite quotient G-graphs Γi and G-maps ϕij : Γi → Γj (i j) over a directed poset (I, ). (b) If G is ﬁnite and acts freely on Γ , then the decomposition of part (a) can be chosen so that G acts freely on each Γi .

Proof The proof follows the same pattern as the proof of Proposition 2.1.4; we only indicate the main steps and changes. We prove (a) and (b) at the same time. Let R be an open equivalence relation on Γ . Assume that G acts continuously on the finite discrete space Γ/R in such a way that the canonical projection ϕR : Γ → Γ/R is a G-map of G-spaces: this is equivalent to saying that whenever m, m ∈ Γ and mR = m R, then (gm)R = (gm )R, for all g ∈ G (we term such R a G-invariant equivalence relation). Then (see Sect. 1.3) there exists a set R of G-invariant open equivalence relations on Γ such that (R, ) is a directed poset, {Γ/R,ϕRR } is an inverse system of finite G-spaces and G-maps over R and = Γ lim←− Γ/R (2.3) R∈R as topological G-spaces. Moreover, if G is finite and acts freely on Γ , one can modify the set R so that the action of G on each Γ/R is free and the decomposition (2.3) still holds. Let R be the subset of R consisting of those R ∈ R such that in addition Γ/R has the structure of a G-graph and ϕR : Γ → Γ/R is a G-map of G-graphs. ∈ R : → Let R and apply Construction 2.1.3 to get the maps ϕR Γ ΓϕR and ψϕ : Γϕ → Γ/R.Forg ∈ G and m ∈ Γ , define R R g ϕ(m),ϕd0(m), ϕd1(m) = gϕ(m), gϕd0(m), gϕd1(m) . 44 2 Profinite Graphs

This makes ΓϕR into a G-graph and one checks that ϕR is a G-map of G-graphs and ˜ ψϕR is a G-map of G-spaces. Let R be the open equivalence relation on Γ whose {−1 | ∈ } = ˜ ˜ equivalence classes are ϕR (x) x ΓϕR , so that ΓϕR Γ/R. Therefore R R. From this one sees, as in the proof of Proposition 2.1.4, that R is a directed poset R that is coﬁnal in . Observe that if G acts freely on Γ/R, then it acts freely on ΓϕR . Hence both (a) and (b) follow from the decomposition (2.3) (see Sect. 1.1).

We remark that part (b) of the above proposition can be sharpened in the following sense. When G is infinite, it obviously cannot act freely on a finite graph; hence, if G acts freely on Γ , it is not possible to obtain a G-decomposition of Γ as in part (a) if in addition one requires that G acts freely on each Γi . However, one can obtain a decomposition as in part (a) so that, for each i, a finite quotient Gi of G acts freely on Γi , and G is the inverse limit of the Gi . We make this precise in Proposition 3.1.3. The following example shows how to do this in the case of Cayley graphs.

Example 2.2.3 (The Cayley graph as a G-graph) Let G be a proﬁnite group and let X be a closed subset of G.LetΓ(G,X)be the Cayley graph of G with respect to X (see Example 2.1.12). Deﬁne a left action of G on Γ(G,X)by setting

g · (g, x) = g g,x ∀x ∈ X˜ = X ∪{1},g,g∈ G.

Clearly gdi(m) = di(gm), for all g ∈ G, m ∈ Γ(G,X), i = 0, 1. Thus, G acts (continuously and freely) on the Cayley graph Γ(G,X). Now, if N is the collection of all open normal subgroups of G,wehave = Γ(G,X) lim←− Γ(G/N,XN ), N∈N where XN is the image of X in G/N. Note that G/N acts freely on Γ(G/N,XN ).

The next lemma sometimes provides a useful way of checking whether certain G-graphs are connected.

Lemma 2.2.4 (a) Let G = X be an abstract group generated by a subset X. Assume that G acts on an abstract graph Γ . Let be a connected subgraph of Γ such that ∩ x = ∅, for all x ∈ X. Then G = g g∈G is a connected subgraph of Γ . (b) Let X be a closed subset of a profinite group G that generates the group topologically, i.e., G = X. Assume that G acts on a profinite graph Γ . Let be a connected profinite subgraph of Γ such that ∩ x = ∅, for all x ∈ X. Then G = g g∈G is a connected profinite subgraph of Γ . 2.3 The Chain Complex of a Graph 45

(c) Let G be a proﬁnite group and let X be a closed subset of G. The Cayley graph Γ(G,X)is connected if and only if G = X.

Proof (a) Put Y = xε ε =±1,x∈ X , and let Yn be the set of elements of G that can be written as a product of not more = = ∞ ⊆ ⊆··· than n elements of Y (n 0, 1, 2,...). Since G n=0 Yn, and Y0 Y1 , it suffices to prove that Yn is a connected graph. We show this by induction on n. If n = 0, then Y0 = . Assume that Yn is connected. From our assumption that x ∩ = ∅, we deduce that x−1 ∩ = ∅, for all x ∈ X. Observe that if w is ε awordinY of length n + 1, then w = w x ,forsomew ∈ Yn and some x ∈ X; hence w ∩ w = ∅; and so, w ∪ Yn is connected. It follows that Yn+1 is connected. (b) By Proposition 2.2.2 there exists a decomposition Γ = lim Γ , where all Γ ←− i i are finite quotient G-graphs of Γ . Hence it suffices to prove the result for Γ finite. In that case the kernel K of the action of G on Γ is an open normal subgroup of G. Therefore, replacing G by its quotient G/K if necessary, we may assume that G is finite; and then the result follows from part (a). (c) Let U be the collection of all open normal subgroups of G. Then = Γ(G,X) lim←− Γ(G/U,XU ), U∈U where XU is the image of X on G/U under the canonical map G → G/U . Therefore we may assume that G is finite, in which case the result follows from part (a): consider the connected subgraph of Γ(G,X) consisting of the vertices 1 and {x | x ∈ X} and the collection of edges {(1,x) | x ∈ X −{1}}; then Γ(G,X)= G.

2.3 The Chain Complex of a Graph

We shall use the following notation and terminology. Given a pseudovariety of finite groups C, we say that R is a pro-C ring if it is an inverse limit of finite rings which are in C as abelian groups; if C is the class of all finite rings, we write profinite rather than pro-C.LetX be a profinite space and let R beapro-C ring. We denote by [[ RX]] the free profinite R-module on the space X. Similarly, [[ R(X,∗)]] denotes the free profinite R-module on a pointed space (X, ∗). The complete group algebra [[ RG]] is the inverse limit of the finite group algebras

[[ RG]] = lim (R/I)(G/U) , ←− where I and U range over the open ideals of R and the open normal subgroups of G, respectively. Let G be a profinite group, and let X be a profinite G-space. Then [[ RX]] naturally becomes a profinite [[ RG]] -module. Similarly, if (X, ∗) is a pointed profinite 46 2 Profinite Graphs

G-space, then the free profinite R-module [[ R(X,∗)]] is naturally a profinite [[ RG]] - module. Let Γ be a profinite graph. Define E∗(Γ ) = Γ/V(Γ) to be the quotient space of the space Γ modulo the subspace of vertices V(Γ).We think of E∗(Γ ) as a pointed space with the image of V(Γ) as the distinguished point. Let R be a profinite ring and consider the free profinite R-modules [[ R(E∗(Γ ), ∗)]] and [[ RV (Γ )]] on the pointed profinite space (E∗(Γ ), ∗) and on the profinite space V(Γ), respectively. Denote by C(Γ,R) the chain complex

∗ d ε 0 −→ R E (Γ ), ∗ −→ RV (Γ ) −→ R −→ 0 (2.4) of free proﬁnite R-modules and continuous R-homomorphisms d and ε determined by ε(v) = 1, for every v ∈ V(Γ), d(e)¯ = d1(e) − d0(e), where e¯ is the image of an edge e ∈ E(Γ ) in the quotient space E∗(Γ ), and d(∗) = 0. Obviously, Im(d) ⊆ Ker(ε). If we need to emphasize the role of the ring R we sometimes write dR for the map d. Note that if E(Γ ) is closed in Γ , then ∗ is an isolated point of E∗(Γ ), and so [[ R(E∗(Γ ), ∗)]] = [[ RE(Γ )]] ; this is the case in many important examples. The homology groups of Γ are deﬁned as the homology groups of the chain complex C(Γ,R) in the usual way:

H0(Γ, R) = Ker(ε)/Im(d), H1(Γ, R) = Ker(d). A qmorphism α : Γ −→ of proﬁnite graphs naturally induces continuous maps

∗ ∗ αV : V(Γ)−→ V() and αE∗ : E (Γ ), ∗ −→ E (), ∗ , which in turn extend to continuous R-homomorphisms

α˜ : RV (Γ ) −→ RV () and V ∗ ∗ α˜ E∗ : R E (Γ ), ∗ −→ R E (), ∗ . Then the following diagram

0 [[ R(E∗(Γ ), ∗)]] d [[ RV (Γ )]] ε R 0

α˜E∗ α˜V idR 0 [[ R(E∗(), ∗)]] d [[ RV ()]] ε R 0 commutes. In other words, the triple α˜ = (α˜ E∗ , α˜V , idR) is a morphism α˜ : C(Γ,R) −→ C(,R) 2.3 The Chain Complex of a Graph 47 of complexes. Therefore, if = Γ lim←− Γi i∈I is an inverse limit of an inverse system of proﬁnite graphs Γi , the corresponding chain complexes C(Γi,R)form an inverse system and = C(Γ,R) lim←− C(Γi,R). i∈I Furthermore, the homomorphism α˜ induces continuous homomorphisms of homology groups ∗ : −→ ∗ : −→ α0 H0(Γ, R) H0(, R) and α1 H1(Γ, R) H1(, R). ∗ ˜ ∗ Of course, α1 is just the restriction of αE to Ker(d). The statements in the following lemma are easily veriﬁed and we leave them to the reader.

Lemma 2.3.1 Let R be a profinite ring. (a) Let α : Γ −→ be a qmorphism of profinite graphs. If α is surjective, then ∗ : −→ α0 H0(Γ, R) H0(, R) is surjective. If α is injective, so is ∗ : −→ α1 H1(Γ, R) H1(, R). (b) If Γ = lim Γ is the inverse limit of an inverse system of profinite graphs Γ , ←− i i then H (Γ, R) = lim H (Γ ,R) and H (Γ, R) = lim H (Γ ,R). 0 ←− 0 i 1 ←− 1 i

In the next proposition we prove that the connectivity of a proﬁnite graph is equivalent to the triviality of its 0-homology group.

Proposition 2.3.2 A proﬁnite graph Γ is connected if and only if H0(Γ, R) = 0, independently of the choice of the proﬁnite ring R.

Proof Write Γ as an inverse limit Γ = lim ∈ Γ of finite quotient graphs Γ .By ←− i I i i Proposition 2.1.6, Γ is a connected profinite graph if and only if each Γi is connected as an abstract graph. On the other hand, by Lemma 2.3.1, H0(Γ, R) = 0if and only if H0(Γi,R)= 0, for each i. Hence it suffices to prove the theorem for finite Γ . In this case the sequence (2.4) becomes

d ε 0 −→ RE(Γ ) −→ RV (Γ ) −→ R −→ 0, 48 2 Proﬁnite Graphs where if X is a set, [RX] denotes the free R-module on the set X. Observe that εd = 0, so that Im(d) ≤ Ker(ε). Assume ﬁrst that Γ is connected. Let t t ε nivi = ni = 0 v1,...,vt ∈ V(Γ); n1,...,nt ∈ R . i=1 i=1

∈ t = t − Fix v0 V(Γ). Then i=1 nivi i=1 ni(vi v0); hence it sufﬁces to check that for every pair of distinct vertices v,w of Γ , there exists some c ∈[RE(Γ )] = − ε1 εm with d(c) w v. To verify this let e1 ,...,em be a path from v to w. Deﬁne = s = − c i=1 εiei , where we think of εi as an element of R. Then d(c) w v. Hence the sequence is exact at [RV (Γ )], i.e., H0(Γ, R) = 0. Assume now that the sequence is exact at [RV (Γ )].Letv ∈ V(Γ) and let Γ be the connected component of v in Γ . Suppose that Γ = Γ , and let Γ be the complement of Γ in Γ ; then Γ is a subgraph of Γ . Choose v ∈ V(Γ ). Clearly v − v ∈ Ker(ε). Then there exists s niei ∈ RE(Γ ) ei ∈ E(Γ ),ni ∈ R,i = 1,...,s i=1

s = − such that d( i=1 niei) v v . We may assume that v is a vertex of e1 and e1,...,et ∈ Γ , while et+1,...,es ∈ Γ and v is a vertex of es . Clearly

d RE Γ ≤ RV Γ ,

d RE Γ ≤ RV Γ and

RV (Γ ) = RV Γ ⊕ RV Γ .

t = ∈ = Therefore d( i=1 niei) v . However, v / Ker(ε), a contradiction. Thus Γ Γ , and Γ is connected.

2.4 π-Trees and C-Trees

Let C be a pseudovariety of ﬁnite groups and consider the set of primes π = π(C) involved in C (see Sect. 1.3). Let ZCˆ denote the pro-C completion of the group of integers Z. This is the free pro-C group of rank 1; it also has, in a natural way, a ring structure. One has np ZCˆ = Zp/p Zp, p∈π where n np = np(C) = sup n n ∈ N,p ||C|,C ∈ C . 2.4 π-Trees and C-Trees 49

∞ If np =∞, then, by convention, we agree that p Zp = 0. Note that every abelian pro-C group is in a unique way a proﬁnite ZCˆ-module. A proﬁnite graph Γ is said to be a C-tree if Γ is connected and H1(Γ, ZCˆ) = 0. Thus Γ is a C-tree if and only if the sequence C(Γ,ZCˆ) (see Sect. 2.3)

∗ d ε 0 −→ ZCˆ E (Γ ), ∗ −→ ZCˆV(Γ) −→ ZCˆ −→ 0 (2.5) is exact. Note that if the set of edges E(Γ ) of Γ is closed, then the sequence (2.5) becomes

−→ d ε 0 ZCˆE(Γ ) −→ ZCˆV(Γ) −→ ZCˆ −→ 0.

Lemma 2.4.1 Let C be a pseudovariety of ﬁnite groups. A proﬁnite graph Γ is a C-tree if and only if the sequence C(Γ,Fp)

∗ d ε 0 −→ Fp E (Γ ), ∗ −→ FpV(Γ) −→ Fp −→ 0 is exact for every p ∈ π(C), where Fp is the ﬁeld with p-elements.

Proof First observe that a proabelian group is the direct product of its p-Sylow subgroups. So, for any profinite space X, np [[ ZCˆX]] = Zp/p Zp X . p∈π(C) Therefore, np C(Γ,ZCˆ) = C Γ,Zp/p Zp , p∈π(C) where np = np(C). Hence the sequence C(Γ,ZCˆ) is exact if and only if the se- n quence C(Γ,Zp/p p Zp) is exact for each p ∈ π(C). Therefore it suffices to prove n that C(Γ,Zp/p p Zp) is exact if and only if C(Γ,Fp) is exact. n We observe that C(Γ,Zp/p p Zp) and C(Γ,Fp) are sequences of free abelian pro-p groups of exponent pnp and free abelian pro-p groups of exponent p, respectively. Moreover, if X is a profinite space, [[ FpX]] is the Frattini quotient

np np Zp/p Zp X /Φ Zp/p Zp X

n of [[ (Zp/p p Zp)X]] : this is obvious if X is finite, and in general this can be deduced by a standard inverse limit argument. n n Exactness of C(Γ,Zp/p p Zp) at [[ (Zp/p p Zp)(V (Γ ))]] is equivalent to exactness of C(Γ,Fp) at [[ Fp(V (Γ ))]] , because any of these statements is equivalent to Γ being connected, according to Proposition 2.3.2. Hence from now on we assume that Γ is connected as a profinite graph, and we must show that injectivity of the n map d of C(Γ,Zp/p p Zp) is equivalent to injectivity of the map d of C(Γ,Fp). 50 2 Profinite Graphs

To prove this we will also work with the chain complex C(Γ,Zp). Consider the commutative diagram

∗ dZp ∗ [[ Zp(E (Γ ), ∗)]] d([[ Zp(E (Γ ), ∗)]] )

n ∗ d n ∗ [[ (Zp/p p Zp)(E (Γ ), ∗)]] d([[ (Zp/p p Zp)(E (Γ ), ∗)]] )

∗ dFp ∗ [[ Fp(E (Γ ), ∗)]] d([[ Fp(E (Γ ), ∗)]] ) where the vertical maps are the natural quotient maps, and the maps dZp , d and dFp n denote the maps induced by the homomorphisms d of C(Γ,Zp), C(Γ,Zp/p p Zp) and C(Γ,Fp), respectively. Since the sequence C(Γ,Zp) is exact at [[ ZpV(Γ)]] and since Zp is the free Zp-module of rank 1, the map ε splits, and we have

∗ ZpV(Γ) = d Zp E (Γ ), ∗ ⊕ Zp. Similarly, we have

np np ∗ np Zp/p Zp V(Γ) = d Zp/p Zp E (Γ ), ∗ ⊕ Zp/p Zp and

∗ FpV(Γ) = d Fp E (Γ ), ∗ ⊕ Fp. From this it follows that the last line of the diagram is obtained from the ﬁrst or second line by taking quotients modulo the subgroups of p-th powers (the Frattini subgroups); and the second line is obtained from the ﬁrst by taking quotients modulo the subgroups of pnp -th powers. It follows that if dZp (respectively, d )isan isomorphism, then so is dFp . Conversely, assume that dFp is an isomorphism. Since ∗ d([[ Zp(E (Γ ), ∗)]] ) is a subgroup of [[ ZpV(Γ)]] , it is a torsion-free pro-p group, and so a free abelian pro-p group (cf. RZ, Theorem 4.3.3 and Example 3.3.8(c)). Therefore there exists a continuous homomorphism

∗ ∗ α : d Zp E (Γ ), ∗ −→ Zp E (Γ ), ∗ Z ∗ such that d p α is the identity map on d([[ Zp(E (Γ ), ∗)]] ); therefore α is injective. On the other hand,

Zp ∗ Ker d ≤ Φ Zp E (Γ ), ∗ and

Zp ∗ Ker d + Im(α) = Zp E (Γ ), ∗ , ∗ ∗ where Φ([[ Zp(E (Γ ), ∗)]] ) is the subgroup of p-th powers of [[ Zp(E (Γ ), ∗)]] , ∗ i.e., its Frattini subgroup. So Im(α) =[[Zp(E (Γ ), ∗)]] (cf. RZ, Corollary 2.8.5).

Therefore α is an isomorphism, and hence dZp is an isomorphism. Thus, d is also an isomorphism. 2.4 π-Trees and C-Trees 51

The above lemma shows that in fact the concept of a C-tree depends only on the primes involved in the pseudovariety C. This suggests the following definition. Let π be a nonempty set of prime numbers, and denote by Zπˆ the profinite group (ring) Zπˆ = Zp. p∈π We say that a profinite graph Γ is a π-tree if it is connected as a profinite graph and one has H1(Γ, Zπˆ ) = 0. In other words, Γ is a π-tree if and only if the sequence C(Γ,Zπˆ )

∗ d ε 0 −→ Zπˆ E (Γ ), ∗ −→ Zπˆ V(Γ) −→ Zπˆ −→ 0 (2.6) is exact. If π ={p} consists of only one prime, we write p-tree rather than {p}- tree. When π is the set of all prime numbers, we normally use the term proﬁnite tree rather than π-tree. The following proposition is an immediate consequence of Lemma 2.4.1.

Proposition 2.4.2 Let C be a pseudovariety of finite groups and let Γ be a profinite graph. Let π = π(C). The following conditions are equivalent: (a) Γ is a C-tree; (b) Γ is a π-tree; (c) let R be a quotient ring of Z such that the order #R of R as a profinite group involves precisely the primes in the set π. Then the sequence

∗ d ε 0 −→ R E (Γ ), ∗ −→ RV (Γ ) −→ R −→ 0 is exact; (d) for a given prime p, let Rp denote one of the following rings: Zp, Fp or n Zp/p Zp, for some positive integer n. Then, for every p ∈ π, the sequence

∗ d ε 0 −→ Rp E (Γ ), ∗ −→ RpV(Γ) −→ Rp −→ 0 is exact.

Proposition 2.4.3 Let π be a nonempty set of prime numbers. Then the following statements hold. (a) Every finite tree is a π-tree. (b) Every connected profinite subgraph of a π-tree is a π-tree. (c) If 1 and 2 are π-subtrees of a π-tree such that 1 ∩ 2 = ∅, then 1 ∪ 2 is a π-subtree. (d) An inverse limit of π-trees is a π-tree. In particular, an inverse limit of finite trees is a π-tree. (e) If ∅=π ⊆ π, then every π-tree is a π -tree.

Proof Part (b) follows from Lemma 2.3.1(a). Part (c) follows from (b) and Lemma 2.1.7. The first statement in part (d) is a consequence of Lemma 2.3.1(b); 52 2 Profinite Graphs and the second then follows from (a). Part (e) is a consequence of the definition of a π-tree. To prove (a), let Γ be a finite tree. In this case the sequence (2.6) becomes

d ε 0 −→ ZpE(Γ ) −→ ZpV(Γ) −→ Zp −→ 0.

Since Γ is connected, this sequence is exact at [ZpV(Γ)] by Proposition 2.3.2.It remains to see that d is an injection. For this deﬁne a map

ρ : V(Γ)−→ ZpE(Γ ) as follows: ﬁx a vertex v0 ∈ V(Γ); since Γ is an abstract tree, for each vertex v ∈ ε1 εt V(Γ)there is a unique path e ,...,et from v0 to v of minimal length; deﬁne 1 ρ(v)= ε1e1 +···+εt et e1,...,et ∈ E(Γ ); εi =±1,i = 1,...,t .

Since [ZpV(Γ)] is a free Zp-module, this map extends to a Zp-homomorphism (also denoted ρ) ρ :[ZpV(Γ)]→[ZpE(Γ )]. Then ρd is the identity map on [ZpE(Γ )]; thus d is an injection.

Exercise 2.4.4 Let T be a π-tree. (a) T does not contain circuits. (b) If v,w ∈ V(T)and there exists a path pvw from v to w, then there is a unique reduced path from v to w.

Example 2.4.5 (A π-tree which is not an inverse limit of finite trees) It is not always possible to decompose a π-tree as an inverse limit of finite trees. For example, let p be a prime number. The Cayley graph Γ = Γ(Zp, {1}) is a p-tree (see Theo- rem 2.5.3 below). Let Γ˜ be a finite quotient graph of Γ . Then Γ˜ is also a quotient graph of a graph of the form Γ(Z/pnZ, {1}) (see Lemma 2.1.5), which is a circuit. Hence, if |Γ˜ |≥2, then Γ˜ is not a tree.

Lemma 2.4.6 Let be a profinite subgraph of a profinite graph Γ , and let R be a profinite ring. Then (a) V()is a closed subspace of V(Γ), and (E∗(), ∗) is naturally embedded in (E∗(Γ ), ∗); (b) V(Γ/) is naturally homeomorphic with V(Γ)/V(), and E∗(Γ /, ∗) is naturally homeomorphic with (E∗(Γ )/E∗(), ∗), where, in this last case, the distinguished point ∗ is the image of E∗() in E∗(Γ )/E∗(); (c) R E∗(Γ /), ∗ =∼ R E∗(Γ ), ∗ R E∗(), ∗ .

Proof Parts (a) and (b) are straightforward. To prove (c) consider the natural continuous map ι : E∗(Γ /), ∗ −→ R E∗(Γ ), ∗ R E∗(), ∗ . 2.4 π-Trees and C-Trees 53

We must show that [[ R(E∗(Γ ), ∗)]] /[[ R(E∗(), ∗)]] is the free proﬁnite R-module on the space (E∗(Γ /), ∗) with respect to the map ι (see Sect. 1.7). Let ϕ : (E∗(Γ /), ∗) → A be a continuous map of pointed spaces into a proﬁnite R- module A. Then ϕ induces a continuous map

∗ ϕ1 : E (Γ ), ∗ −→ A, and this in turn induces a continuous R-homomorphism

∗ ϕ1 : R E (Γ ), ∗ −→ A ∗ such that ϕ1([[ R(E (), ∗)]] ) = 0. Therefore ϕ1 induces a continuous R-homomorphism ϕ¯ : R E∗(Γ ), ∗ R E∗(), ∗ −→ A such that ϕι¯ = ϕ. The uniqueness of ϕ¯ is clear since ι(E∗(Γ /), ∗) generates [[ R(E∗(Γ ), ∗)]] /[[ R(E∗(), ∗)]] .

Lemma 2.4.7 Let be a π-subtree of a connected proﬁnite graph Γ and let α : Γ −→ Γ/ be the corresponding canonical epimorphism of graphs. Then the induced homomorphism ∗ : −→ α1 H1(Γ, Zπˆ ) H1(Γ /, Zπˆ ) is an isomorphism. In particular, if Γ is a π-tree, then so is Γ/.

Proof We may assume that π consists of just one prime p.Let β : −→ Γ ˜ be the natural embedding. Then β and α induce a monomorphism β : C(,Zp) → C(Γ,Zp) and an epimorphism α˜ : C(Γ,Zp) → C(Γ/,Zp) of chain complexes, respectively, and the following diagram 0

∗ d ε 0 [[ Zp(E (), ∗)]] [[ ZpV()]] Zp 0 ˜ ˜ βE∗ βV id ∗ dΓ εΓ [[ Zp(E (Γ ), ∗)]] [[ ZpV(Γ)]] Zp 0

α˜E∗ α˜V id ∗ dΓ/ εΓ/ [[ Zp(E (Γ /), ∗)]] [[ ZpV(Γ/)]] Zp 0

0 54 2 Profinite Graphs commutes. Note that the first row is exact because is a p-tree, the second row is exact because Γ is connected. ˜ ∗ By Lemma 2.4.6,Ker(α˜E∗ ) = βE∗ ([[ Zp(E (), ∗)]] ),inotherwords,thefirst column of the diagram is an exact sequence. From this it easily follows that ∗ ∈ ∗ = α1 is an injection. Indeed, let a H1(Γ, Zp) be such that α1 (a) 0; i.e., ∗ Γ a ∈[[Zp(E (Γ ), ∗)]] with d (a) = 0 and α˜ E∗ (a) = 0. Then there exists a b ∈ ∗ ˜ ¯ [[ Zp(E (), ∗)]] such that βE∗ (b) = a.Now,sinced and βV are injections, we deduce from the commutativity of the diagram that b = 0. Thus a = 0. ˜ Next we observe that Ker(α˜ V ) = βV (Ker(ε )); indeed, first we notice that this is straightforward if V(Γ)is finite; in general we use an inverse limit argument. ∗ ∈[[ ∗ ∗ ]] Now we can easily deduce that α1 is a surjection: if c Zp(E (Γ /), ) Γ/ ∗ Γ and d (c) = 0, choose c˜ ∈[[Zp(E (Γ ), ∗)]] such that α˜ E∗ (c)˜ = c; then d (c)˜ ∈ ˜ Γ Ker(α˜ V ), and so there exists a y ∈ Ker(ε ) with βV (y) = d (c)˜ ; hence there exists ∗ ˜ Γ a y ∈[[Zp(E (), ∗)]] with d (y ) = y; then c =˜c − βE∗ (y ) ∈ Ker(d ) and

α˜ E∗ (c ) = c, as needed.

Lemma 2.4.8 Let R be a profinite ring. Then the following statements hold. { | ∈ } (a) Let Xi i I be a collection of closed subspaces of a profinite space Y . Set = X i∈I Xi . Then [[ RX]] = [[ RXi]] . i { ∗ | ∈ } (b) Let (Xi, ) i I be a collection of closed pointed subspaces of a profinite ∗ ∗ = ∗ pointed space (Y, ). Set (X, ) i∈I (Xi, ). Then R(X,∗) = R(Xi, ∗) . i (c) Let Y and Z be closed subspaces of the profinite pointed space (X, ∗) such that ∗∈Y and ∗ ∈/ Z. Then there are natural isomorphisms

R(X,∗) /[[ RZ]] =∼ R(X/Z,∗) and R(X,∗) R(Y,∗) =∼ R(X/Y,∗) .

Proof The proofs of (a) and (b) are similar. We only prove (a). Assume first that I ={1, 2}, i.e., X = X1 ∩ X2. Write Y as the inverse limit = Y lim←− Yj j∈J of its finite quotient spaces. Denote by ϕj : Y → Yj the projection (j ∈ J), and define X1j = ϕj (X1) and X2j = ϕj (X2). Since X1j and X2j are finite, we have

R(X1j ∩ X2j ) =[[RX1j ]] ∩ [[ RX2j ]] . 2.4 π-Trees and C-Trees 55

It is easy to verify that ∩ = ∩ = ∩ X1 X2 lim←− X1j lim←− X2j lim←− (X1j X2j ). j∈J j∈J j∈J Hence ∩ = ∩ = ∩ R(X1 X2) R lim←− (X1j X2j ) lim←− R(X1j X2j ) j∈J j∈J = [[ ]] ∩ [[ ]] = [[ ]] ∩ [[ ]] lim←− RX1j RX2j lim←− RX1j lim←− RX2j j∈J j∈J j∈J =[[RX1]] ∩ [[ RX2]] (for the second and fourth equalities see RZ, Proposition 5.2.2). Assume now that I is any indexing set. By the case considered above we may assume that the collection {Xi | i ∈ I} is ﬁltered from below, i.e., that the intersection of any two sets in the collection contains a set in the collection. So we may think of this collection as an inverse system of sets and = = X Xi lim←− Xi. i∈I i∈I

Also, using again the case above, the collection of profinite R-submodules {[[RXi]] | i ∈ I} of [[ RY]] is filtered from below. Therefore, [[ ]] = = [[ ]] = [[ ]] RX R lim←− Xi lim←− RXi RXi . i∈I i∈I i∈I (c) We prove the second statement, the first being similar. The quotient map (X, ∗) → (X/Y, ∗) induces a continuous epimorphism of free profinite modules f :[[R(X,∗)]] → [[ R(X/Y,∗)]] . Since f([[ R(Y,∗)]] ) = 0, f induces an epimorphism ρ : R(X,∗) R(Y,∗) −→ R(X/Y,∗) . On the other hand, the natural map (X/Y, ∗) →[[R(X,∗)]] /[[ R(Y,∗)]] induces a continuous homomorphism ψ : R(X/Y,∗) −→ R(X,∗) R(Y,∗) . Finally, observe that the composition ψρ is the identity map on [[ R(X,∗)]] / [[ R(Y,∗)]] . Thus ρ is an isomorphism.

Proposition 2.4.9 Let π be a nonempty set of prime numbers . Suppose that {i | ∈ } = i I is a family of π-subtrees of a π-tree T , and let i∈I i . Then is either empty or a π-tree.

Proof Assume that = ∅. By Lemma 2.4.8 one has Zπˆ V() = Zπˆ V(i) i∈I 56 2 Proﬁnite Graphs and ∗ ∗ Zπˆ E (), ∗ = Zπˆ E (i), ∗ . i∈I Consider the exact sequence

∗ d ε 0 −→ Zπˆ E (T ), ∗ −→ Zπˆ V(T) −→ Zπˆ −→ 0 associated with T . Denote by ε,εi ,d,di the restrictions of ε and d to and i , respectively. Then i Ker ε = Zπˆ V() ∩ Ker(ε) = Zπˆ V(i) ∩ Ker(ε) = Ker ε ; i∈I i∈I moreover, Im d = Im di i∈I because d is injective. Since each i is connected, we have Ker(ε i ) = Im(d i ), for every i, by Proposition 2.3.2. It follows that Im(d) = Ker(ε). So, by Proposition 2.3.2, is connected, and therefore a π-tree according to Proposi- tion 2.4.3(b).

It follows from Proposition 2.4.9 that given a nonempty subset W of a π-tree T , there exists a smallest π-subtree [W] containing W , namely the intersection of all π-subtrees containing W .IfW consists of two vertices v and w, we use the notation [v,w] rather than [{v,w}] and call it the chain connecting v and w. Observe that if [v,w] is ﬁnite, then it is just the underlying graph of the unique reduced path from v to w.

Lemma 2.4.10 A proﬁnite subgraph of a π-tree T is a π-tree if and only if [v,w]⊆, for all v,w ∈ V().

Proof If is a π-tree, then by definition [v,w]⊆, for all v,w ∈ V(). Con- versely, suppose is a profinite subgraph of T and that [v,w]⊆, for all v,w ∈ V(). To prove that is a π-tree, it suffices to show that is connected (see Proposition 2.4.3(b)). Write T as an inverse limit of finite quotient graphs, = T lim←− Ti, i∈I and let ϕi : T → Ti denote the projection (i ∈ I ). It suffices to prove that ϕi() is a connected graph for each i ∈ I . Given vertices v¯ and w¯ of ϕi(),letv,w ∈ V() with ϕi(v) =¯v and ϕi(w) =¯w. Since [v,w]⊆ and [v,w] is a π-tree, we have that ϕi([v,w]) is a connected subgraph of the finite graph ϕi() containing v¯ and w¯ . Therefore, ϕi() is connected. 2.5 Cayley Graphs and C-Trees 57

Example 2.4.11 (A π-tree that coincides with its infinite chains) Let Γ = Γ(Z, 1) be the Cayley graph of the free profinite group Z of rank 1 with respect to its subset {1}.Thisisaπ-tree for any nonempty set of prime numbers π (see Theorem 2.5.3 below and Proposition 2.4.3(e)). The proper π-subtrees of Γ are precisely the proper connected profinite subgraphs of Γ , and these are precisely the finite π-subtrees (see Example 2.1.13). Therefore, if v,w are vertices of Γ , then [v,w]=Γ , unless [v,w] is finite, in which case [v,w] has vertices γ,γ + 1,...,γ + t, where γ = v or γ = w and t is a natural number.

Let G be a proﬁnite group that acts on a π-tree T .Aπ-subtree T of T is G- invariant if whenever g ∈ G and m ∈ T , one has gm ∈ T ; and such T is minimal if it does not contain any proper G-invariant π-subtrees. Minimal G-invariant π- subtrees are especially useful when they are unique. In the next proposition we begin the study of minimal G-invariant π-subtrees T of T . A more systematic study is carried out in Chap. 8.

Proposition 2.4.12 Let G be a profinite group acting on a π-tree T . Then the following assertions hold. (a) There exists a minimal G-invariant π-subtree D of T . (b) If |D| > 1, then D is unique. In particular, if |G| > 1 and G acts freely on T or if G is infinite and the stabilizer of some m ∈ D is finite, then D is the unique minimal G-invariant π-subtree of T . (c) Assume that D is unique. Let N G be such that there exists a unique minimal N-invariant π-subtree L of T . Then L = D.

Proof (a) Consider the collection T of all G-invariant π-subtrees of T ordered by ∈ T T = ∅ { } T reverse inclusion. Since T , .Let Ti i∈I be a linearly ordered chain in . By the compactness of T ,theset Ti is nonempty. Then, by Proposition 2.4.9, Ti is a G-invariant π-subtree. So {Ti}i∈I possesses an upper bound in T . Therefore we can apply Zorn’s lemma to conclude that there exists a minimal G-invariant π- subtree. (b) This will be proved after Corollary 4.1.9. (c) Let g ∈ G; then N acts on gL and so gL is minimal N-invariant; hence gL = L. This means that G acts on L. Therefore D ⊆ L; but obviously L ⊆ D, since N acts on D; thus L = D.

2.5 Cayley Graphs and C-Trees

A pseudovariety of finite groups C0 is said to be closed under extensions with abelian kernel if whenever 1 −→ A −→ G −→ H −→ 1 58 2 Profinite Graphs is an exact sequence of finite groups with A,H ∈ C0 and A is abelian, then G ∈ C0. By the KaloujnineÐKrassner theorem (cf. Kargapolov and Merzljakov 1979, The- orem 6.2.8) such an extension group G can be embedded in the wreath product A by H ; it follows that to check that a pseudovariety of finite groups C is closed under extensions with abelian kernel, it suffices to verify that any semidirect product of an abelian group in C by a group in C is in C. Next we give an example showing that a pseudovariety which is closed under extensions with abelian kernel is not necessarily extension-closed.

Example 2.5.1 (A pseudovariety closed under extensions with abelian kernel that is not extension-closed) Let = A5 be the alternating group of degree 5. This is the finite simple nonabelian group with smallest order. Let C() be the collection of all the finite direct products of copies of . Observe that C() is closed under homomorphic images (cf. RZ, Lemma 8.2.4). For a finite group G, denote by S(G) its maximal solvable normal subgroup. Define V to be the set of all finite groups G such that G/S(G) ∈ C(). We shall show that V is a pseudovariety of finite groups that is closed under extensions with abelian kernel, but not extension-closed. We claim first that V is a pseudovariety. Clearly V is closed under finite direct products; moreover, since C() is closed under homomorphic images, so is V.It remains to prove that V is closed under taking subgroups. Let G ∈ V and let H be a proper subgroup of G. We use induction on the order of G to show that H ∈ V. If G is solvable or G =∼ , then H is solvable and the result is clear. Observe that H/S(H) is a quotient of H/H ∩ S(G).IfS(G) = 1, the result follows from the induction hypothesis since H/H ∩ S(G) =∼ H S(G)/S(G) ≤ G/S(G) and G/S(G) < |G|.

Thus from now on we may assume that G ∈ C() , i.e., G = 1 ×···×n (n ≥ 2), where each i is isomorphic to . Since H is a proper subgroup of G, there is some i such that Hi = H ∩ i = i ,1≤ i ≤ n. Then Hi is solvable. So Hi ≤ S(H) and S(H/Hi) = S(H)/Hi . Now, since H/Hi ≤ G/i ∈ V, we conclude from the induction hypothesis that

H/S(H)= (H/Hi)/S(H/Hi) ∈ C(). This proves the claim. It follows easily from the deﬁnition that V is closed under extensions with abelian kernel. Let us show now that it is not extension-closed. For this consider the wreath product R = C of with a group C of order 2; this is a semidirect product of B = × by C; both of these groups are in V; and the action of C on B permutes the two factors .LetK R and assume that K is solvable. We claim that K = 1. Note that K ∩ B = 1, for otherwise K must contain one of the copies of , contradicting the solvability of K.IfK = 1, we have R = B × K = B × C, contradicting the deﬁnition of R. This proves the claim. Therefore S(R) = 1. Finally, observe that R/∈ C(). Thus R/∈ V. 2.5 Cayley Graphs and C-Trees 59

If (X, ∗) is a pointed proﬁnite space, we denote by F = FC(X, ∗) the free pro-C group on the pointed space (X, ∗). The next two results establish conditions under which the Cayley graph of a free pro-C group with respect to one of its bases is a C-tree. We begin with a study of the augmentation ideal (see Sect. 1.10)ofafree pro-C group.

Lemma 2.5.2 Let C be a pseudovariety of finite groups. Then C is closed under extensions with abelian kernel if and only if for every pointed profinite space (Y, ∗), the augmentation ideal (( I F )) of the complete group algebra [[ ZCˆF ]] of the free pro-C group F = FC(Y, ∗) is a free [[ ZCˆF ]] -module on the pointed space (Y, ∗) with respect to the map ι : (Y, ∗) → (( I F )) defined by ι(y) = y − 1 (y ∈ Y).

Proof The augmentation ideal (( I F )) is topologically generated by the space Y − 1 ={y − 1 | y ∈ Y } as an [[ ZCˆF ]] -module (see Sect. 1.10). Assume first that C is closed under extensions with abelian kernel. We shall prove that (( I F )) satisfies the required universal property of a free [[ ZCˆF ]] -module with respect to the map ι. We must prove that given a map of pointed spaces ψ : Y → M to a profinite [[ ZCˆF ]] -module M, there exists a unique continuous [[ ZCˆF ]] -module homomorphism ψ˜ : (( I F )) → M such that ψι˜ = ψ.

ψ˜ y − 1 (( I F )) M

ι ψ y Y Observe that if such a ψ˜ exists, then it is unique since ι(Y) generates (( I F )) as a [[ ZCˆF ]] -module. We may assume that M is finite since M is an inverse limit of finite [[ ZCˆF ]] - modules (see Sect. 1.7). Note that M ∈ C since M is automatically a ZCˆ-module and so an abelian pro-C group. Since M is in particular an F -module, we may construct the corresponding semidirect product M F . We remark that M F is a pro-C group since C is closed under extensions with abelian kernel. Since F is a free pro-C group on (Y, ∗), there exists a unique continuous homomorphism ρ : F −→ M F such that ρ(y)= (ψ(y), y) (y ∈ Y ). Define now a map δ : F −→ M by the equation (δ(f ), f ) = ρ(f), for all f ∈ F . Then δ is continuous and it is a derivation, that is,

δ(f1f2) = δ(f1) + f1δ(f2), ∀f1,f2 ∈ F (see Sect. 1.10). Now, (see 1.10.7 in Sect. 1.10), there exists an isomorphism

∼ = [[ ]] Der(F, M) Hom ZCˆF (( I F )) , M , 60 2 Profinite Graphs and under this isomorphism δ corresponds to a [[ ZCˆF ]] -homomorphism ψ˜ : (( I F )) −→ M such that ψ(f˜ − 1) = δ(f), for all f ∈ F . Then ψι(y)˜ = ψ(y˜ − 1) = δ(y) = ψ(y), ∀y ∈ Y, and thus ψι˜ = ψ. Conversely, assume that (( I F )) is a free [[ ZCˆF ]] -module on the pointed space (Y, ∗) with respect to the map ι, for every profinite pointed space (Y, ∗), where F = F(Y,∗) denotes the free pro-C group on the pointed profinite space (Y, ∗).Let A,H ∈ C, with A abelian. Assume that A is an H -module, and let G = A H be the corresponding semidirect product. To prove that C is closed under extensions with abelian kernel it suffices to show that G ∈ C, as pointed out above. Let {(ay,hy) | y ∈ Y } be a generating set of G = A H , with ay ∈ A,hy ∈ H , for all y ∈ Y , where (Y, ∗) is a certain finite pointed indexing set and a∗ = 1, h∗ = 1. Then H = hy | y ∈ Y .LetF = FC(Y, ∗) be the free pro-C group on the pointed space (Y, ∗) and let ϕ : F −→ H be the continuous epimorphism determined by ϕ(y) = hy (y ∈ Y ). Then the action of H on A induces an action of F on A via ϕ: f · a = ϕ(f)a, (a ∈ A,f ∈ F). Let G˜ = A F be the corresponding semidirect product, and let ϕ˜ : G˜ = A F −→ G = A H be the epimorphism induced by ϕ. Since, by assumption, (( I F )) is a free [[ ZCˆF ]] -module on (Y, ∗) and A is an [[ ZCˆF ]] -module, there exists a continuous [[ ZCˆF ]] -homomorphism ψ˜ : (( I F )) → A ˜ such that ψ(y − 1) = ay (y ∈ Y). Define d : F −→ A by d(f) = ψ(f˜ − 1)(f∈ F). Then d is a continuous derivation (see 1.10.7 in Sect. 1.10). Hence the map ρ : F −→ G˜ = A F given by ρ(f) = (d(f ), f ) (f ∈ F), is a continuous homomorphism (cf. RZ, Lemma 9.3.6). Define α : F → G to be the composite α =˜ϕρ. Observe that

α(y) = (ay,hy)(y∈ Y); therefore α is an epimorphism, and thus G ∈ C, as needed. 2.5 Cayley Graphs and C-Trees 61

Theorem 2.5.3 Let C be a pseudovariety of ﬁnite groups. Then C is closed under extensions with abelian kernel if and only if for every proﬁnite pointed space (Y, ∗), the Cayley graph Γ = Γ(F,Y)ofthefreepro-C group F = F(Y,∗) with respect to Y is a C-tree.

Proof We think of (Y, ∗) as being embedded in F ; in particular ∗ is identified with 1. Since 1 ∈ Y , Γ = Γ(F,Y)= F × Y and V(Γ)= F ×{1}. Consider the sequence associated with the graph Γ and ZCˆ as in Eq. (2.4) of Sect. 2.3: d ε 0 −→ ZCˆ (F × Y)/ F ×{1} , ∗ −→ [[ ZCˆF ]] −→ ZCˆ −→ 0, where d(f,y) = fy − f (y ∈ Y ) and ε(f ) = 1(f ∈ F ). We have to prove that this sequence is exact for every (Y, ∗) if and only if C is closed under extensions with abelian kernel. By Lemma 2.2.4, Γ is a connected profinite graph since F is topologically generated by Y . Therefore, by Proposition 2.3.2, the above sequence is exact at [[ ZCˆF ]] . It remains to prove that d is a monomorphism. Now, Ker(ε) is the augmentation ideal (( I F )) of [[ ZCˆF ]] , which is generated as a topological [[ ZCˆF ]] -module by the subspace {y − 1 | y ∈ Y } (see Sect. 1.10). On the other hand, [[ ZCˆ((F × Y)/(F ×{1}), ∗)]] is a free [[ ZCˆF ]] -module on the quotient space F \((F × Y)/(F ×{1}), ∗) (cf. RZ, Proposition 5.7.1). The space F \((F × Y)/(F ×{1}), ∗) can be identified with the pointed space ({(1,y)| y ∈ Y }, ∗). Since d(1,y)= 1 − y(y∈ Y), to show that d is a monomorphism is equivalent to showing that the augmentation ideal (( I F )) is free on the subspace ({1 − y | y ∈ Y }, ∗), as a profinite [[ ZCˆF ]] -module. But, according to Lemma 2.5.2, this is the case if and only if C is closed under extensions with abelian kernel. Chapter 3 The Fundamental Group of a Profinite Graph

3.1 Galois Coverings

If a profinite group G acts freely on a profinite graph Γ , then the natural epimorphism of profinite graphs ζ : Γ → = G\Γ of Γ onto the quotient graph = G\Γ is called a Galois covering of the profinite graph . Observe that a Galois covering ζ always sends edges to edges. We shall refer to the group G as the group associated with the Galois covering ζ and denote it by G = G(Γ |) or by G = G(ζ ).IfΓ is finite, one says that the Galois covering ζ is finite (observe that if this is the case, then the associated group G(Γ |) is finite). The Galois covering ζ is said to be connected if Γ is connected.

Example 3.1.1 (a) Let ζ : Γ → be a Galois covering of the profinite graph with associated group G = G(Γ |).LetK c G. Then G/K acts freely on K\Γ and K\Γ → is also a Galois covering of , with associated group G/K. (b) Let (X, ∗) be a pointed profinite space. Define a profinite graph B = B(X,∗) by B = X, V(B)={∗}, di(x) =∗(x ∈ X),fori = 0, 1. We shall refer to B(X,∗) as the bouquet of loops associated to (X, ∗). For example, if X has 5 points, B(X,∗) is the graph

•

Let G be a proﬁnite group, X a closed subset of G and let Γ(G,X) be the Cayley graph of G with respect to X. Then the natural action of G on Γ(G,X) described in Example 2.2.3 is free. Hence the natural epimorphism

ζ : Γ(G,X)→ G\Γ(G,X)

© Springer International Publishing AG 2017 63 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_3 64 3 The Fundamental Group of a Proﬁnite Graph

is a Galois covering. Note that G\Γ(G,X) is just the bouquet of loops B(X ∪{1}, 1).

Let ζ1 : Γ1 → 1, ζ2 : Γ2 → 2 be Galois coverings and put G1 = G(Γ1|1) and G2 = G(Γ2|2).Amorphism of Galois coverings

ν : ζ1 → ζ2 consists of a pair ν = (γ, f ), where γ : Γ1 → Γ2 is a qmorphism of proﬁnite graphs and f : G1 → G2 is a continuous homomorphism of groups such that γ(gm)= f(g)γ(m), for all g ∈ G1, m ∈ Γ1 (we also refer to this situation by saying that the qmorphism of graphs γ and the group homomorphism f are compatible or that the map γ is equivariant). The composition of morphisms of Galois coverings is again a morphism, and Galois coverings and their morphisms form a category. Note that such a morphism ν of Galois coverings induces a unique qmorphism of proﬁnite graphs δ : 1 → 2 such that the diagram

γ Γ1 Γ2

ζ1 ζ2 δ 1 2 commutes. The morphism ν = (γ, f ) is called surjective or an epimorphism if γ and f (and hence δ) are epimorphisms. Assume that {ζi : Γi → i,νij = (γij ,fij ), I} is an inverse system of Galois coverings indexed by a directed poset I . Put Gi = G(ζi)(i∈ I). Then we have corresponding inverse systems {Γi,γij ,I} and {Gi,fij ,I} of profinite graphs and profinite groups, respectively, such that the profinite group

G = lim G ←− i acts continuously and freely on the proﬁnite graph

Γ = lim Γ ; ←− i moreover, the quotient proﬁnite graph G\Γ is isomorphic with the proﬁnite graph = lim . Hence we have established the following proposition. ←− i

Proposition 3.1.2 The inverse limit lim ζ of Galois coverings ←− i

ζi : Γi → i with associated group Gi = G(ζi) is a Galois covering with associated group lim G . ←− i 3.1 Galois Coverings 65

Proposition 3.1.3 Any Galois covering ζ : Γ → of proﬁnite graphs can be de- composed as an inverse limit of ﬁnite Galois coverings, with surjective projections.

Proof Put G = G(Γ |). Assume first that G is finite. By Proposition 2.2.2(b), there exists a G-decomposition = Γ lim←− Γi i∈I of Γ as an inverse limit of finite quotient graphs Γi of Γ on which G acts freely. Denote by

ζi : Γi −→ G\Γi the natural morphism of graphs. Then each ζi is a Galois covering with associated group G. Moreover, the ζi constitute an inverse system of Galois coverings and = ζ lim←− ζi i∈I is the desired decomposition, with surjective projections ζ → ζi . Now assume that G is infinite. Let N denote the set of all open normal subgroups of G. For every N ∈ N , the finite quotient group G/N acts freely on ΓN = N\Γ (see Lemma 2.2.1). By the finite case above, there exists a G/N-decomposition = ΓN lim←− ΓN,i, i∈IN where each ΓN,i is a finite quotient graph of ΓN with free action of G/N on each ΓN,i. Denote by

ϕN,i : ΓN −→ ΓN,i the canonical projection; note that each ϕN,i is surjective. Next we show that the ﬁnite graphs ΓN,i form in a natural way an inverse system whose maps (to be deﬁned presently)

γ(M,j),(N,i) : ΓM,j −→ ΓN,i are compatible with the canonical homomorphisms G/M → G/N, whenever M,N ∈ N and M ≤ N; i.e., we shall show that we have an inverse system of ﬁ- nite Galois coverings. First we specify the poset I over which this is an inverse system. We relabel the elements of the indexing set IN : an element i ∈ IN will be denoted from now on by (N, i). Deﬁne the indexing set of this inverse system to be I = IN . N∈N If (N, i), (M, j) ∈ I , we say that

(N, i) (M, j) 66 3 The Fundamental Group of a Proﬁnite Graph if M ≤ N and there exists a qmorphism of graphs α : ΓM,j → ΓN,i, compatible with the canonical homomorphism G/M → G/N, such that the diagram

ϕN,i N\Γ = ΓN ΓN,i

Γ α

M\Γ = ΓM ΓM,j ϕM,j commutes. Observe that α is unique, if it exists, because ϕM,j is surjective. Hence, (I, ) is a partially ordered set. We observe that the restriction of to IN coincides with the partial ordering in IN (N ∈ N ). We claim that this ordering makes (I, ) into a directed poset. To see this, consider (N, i), (M, j) ∈ I . Put L = N ∩ M. Then (see Lemma 2.1.5), there exists some (L, k) ∈ IL and morphisms of graphs αi : ΓL,k → ΓN,i and αj : ΓL,k → ΓM,j such that the diagram

ΓN ΓN,i

αi

ΓL ΓL,k αj

ΓM ΓM,j commutes. All maps in this diagram are epimorphisms; it follows that αi and αj are unique. All maps except possibly αi and αj are compatible with the natural homomorphisms of the relevant groups acting on these graphs. One deduces that the qmorphisms αi and αj are compatible with the homomorphisms G/L → G/N and G/L → G/M, respectively. This shows that (I, ) is directed. For (N, i) ∈ I , consider the Galois covering

ζN,i : ΓN,i → (G/N)\ΓN,i with associated group G/N. Then one checks that = ζ lim←− ζN,i, N,i∈I with epimorphic projections ζ → ζN,i. 3.1 Galois Coverings 67

The following proposition gives an equivalent way of viewing morphisms of Ga- lois coverings.

Proposition 3.1.4 Let ζ1 : Γ1 → 1, ζ2 : Γ2 → 2 be connected Galois coverings with associated groups G1 = G(Γ1|1) and G2 = G(Γ2|2), respectively. Let γ : Γ1 → Γ2 and δ : 1 → 2 be morphisms of graphs such that the diagram

γ Γ1 Γ2

ζ1 ζ2 δ 1 2 commutes. Then there exists a unique continuous homomorphism

f : G1 −→ G2 which is compatible with γ . Explicitly, f is deﬁned as follows: given g ∈ G1, choose any m ∈ Γ1; then f(g)is the unique element of G2 such that γ(gm)= f(g)γ(m).

Proof Fix g ∈ G1. The commutativity of the diagram implies that γ(gm)and γ(m) are in the same G2-orbit for any given m ∈ Γ1. Since G2 acts freely on Γ2, there exists a unique element h(m, g) ∈ G2 such that γ(gm)= h(m, g)γ (m). We claim that the element h(m, g) is independent of the choice of m and that the map f : G1 → G2 given by g → h(m, g) is a continuous homomorphism. To prove the claim we consider two cases.

Case 1. Assume that ζ2 : Γ2 → 2 is ﬁnite.

For k ∈ G2,set Γ1(k) = m ∈ Γ1 h(m, g) = k .

Observe that Γ1(k) is a closed subset of Γ1; indeed, if m ∈ Γ1 is such that γ(gm)= kγ(m), then, by the continuity of γ and of the actions, there exists some open neighbourhood U of m such that γ(gU)∩ kγ(U) =∅; hence U ∩ Γ1(k) =∅, showing that the complement of Γ1(k) is open; therefore Γ1(k) is closed. Since G2 is ﬁnite, the graph Γ1 is a ﬁnite union of subsets Γ1(k) with pairwise empty intersection. Let us check that Γ1(k) is a subgraph for any k ∈ G2. Assume that m ∈ Γ1(k).This means that γ(gm)= kγ(m); hence

γ gdj (m) = γ dj (gm) = dj γ(gm) = dj kγ(m) = kγ dj (m) , so that dj (m) ∈ Γ1(k) (j = 0, 1). Then by the connectedness of Γ1, Γ1 = Γ1(k0), for some k0 ∈ G2 (see Lemma 2.1.9), i.e., h(m, g) = k0 for every m ∈ Γ1. We are now in a position to deﬁne the homomorphism f .Forg ∈ G1,set

f(g)= h(m, g) = k0. 68 3 The Fundamental Group of a Proﬁnite Graph

The above considerations show that f is well-deﬁned. The equality

f(g1g2)γ (m) = γ(g1g2m) = f(g1)γ (g2m) = f(g1)f (g2)γ (m) for g1,g2 ∈ G1 implies that f is a homomorphism. It remains to show that f is continuous. To see this we use Proposition 3.1.3 to decompose the Galois covering ζ1 : G1 → 1 as an inverse limit

= ζ1 lim←− ζ1i i∈I of ﬁnite Galois coverings ζ1i : Γ1i → 1i with epimorphic projections (γ1i,f1i) : ζ1 → ζ1i . Denote by δ1i : 1 → 1i the induced qmorphism of proﬁnite graphs.

γ Γ1 Γ2 γ1i

Γ1i ζ1 ζ2

δ ζ1i 1 2 δ1i

By Lemma 2.1.5, γ factors through Γ1k,forsomek ∈ I ,sayγk : Γ1k → Γ2 is a qmorphism of proﬁnite graphs such that γ = γkγ1k. Similarly, there exists a morphism δj : 1j → 2 such that δ = δj δ1j . Choose t ∈ I such that t j,k. Since γ1t is surjective, we deduce that ζ2γt = δt ζ1t . It follows from the construction above that there exists a unique map of ﬁnite groups

ft : G1t = G(Γ1t |1t ) −→ G2 such that γi(g1t m1t ) = ft (g1t )γt (m1t ), for any g1t ∈ G1t and any m1t ∈ Γ1t .Nowit follows from the uniqueness of f and ft that f = ft f1t . Since both ft and f1t are continuous, so is f , as desired.

Case 2. Assume that Γ2 is inﬁnite.

By Proposition 3.1.3, one can represent ζ2 as an inverse limit

= ζ2 lim←− ζ2i i∈I 3.1 Galois Coverings 69 of ﬁnite quotient Galois coverings ζ2i : Γ2i → 2i with ﬁnite associated groups G2i = G(Γ2i|2i), where = G2 lim←− G2i. i∈I

Denote by ϕ2i = (γ2i,f2i) : ζ2 → ζ2i the projection. Then we have commutative diagrams

γ γ2i Γ1 Γ2 Γ2i

ζ1 ζ2 ζ2i

δ δ2i 1 2 2i where δ2i is induced by ϕ2i . By the case above, for each i ∈ I there exists a unique continuous group homomorphism fi : G → G2i , compatible with γ2iγ , corresponding to the commutative diagram

γ2i γ Γ1 Γ2i

ζ1 ζ2i

δ2i δ 1 2i

The uniqueness of the homomorphisms fi implies that fi = fjifj , whenever i j, where fji : G2j → G2i is the canonical homomorphism associated with the inverse system {G2i}i∈I . Therefore, the fi induce a unique continuous homomorphism : −→ = f G1 G2 lim←− G2i i∈I compatible with γ .

It follows from this proposition that a morphism ν = (γ, f ) : ζ1 → ζ2 of connected Galois coverings ζ1 : Γ1 → 1, ζ2 : Γ2 → 2 can be viewed equivalently as a pair of morphisms of proﬁnite graphs γ : Γ1 → Γ2 and δ : 1 → 2 such that ζ2γ = δζ1.

Proposition 3.1.5 Let ζ : Γ → , ξ : Σ → be connected Galois coverings and let γ : Γ → Σ be a qmorphism such that ζ = ξγ.

γ Γ Σ

ζ ξ 70 3 The Fundamental Group of a Proﬁnite Graph

Let f : G(Γ |) → G(Σ|) be the homomorphism constructed in Proposi- tion 3.1.4. Then (a) the maps γ and f are surjective; (b) γ is a Galois covering with G(Γ |Σ)= Ker(f ), and consequently G(Σ|) =∼ G(Γ |)/G(Γ |Σ).

Proof (a) First we show that the surjectivity of γ implies the surjectivity of f .Let k ∈ G(Σ|). Choose m ∈ Γ . Since γ is assumed to be onto, there exists some n ∈ Γ such that kγ(m) = γ(n). Clearly ζ(m)= ζ(n). Hence there exists some g ∈ G(Γ |) with gm = n. It follows from the definition of f that γ(gm)= f(g)γ(m); therefore, f(g)γ(m)= kγ(m). Thus f(g)= k, because G(Σ|) acts freely on Σ. Next we prove that γ is surjective. For any open normal subgroup N of G(Σ|), the finite quotient group G(Σ|)/N acts freely on the quotient graph N\Σ.So the natural qmorphism ξN : N\Σ → is a Galois covering with finite associated group G(Σ|)/N. Clearly ξ = lim ξ and the projections of this decomposition ←− N are surjective. Let μN : Σ → N\Σ be the natural morphism and set γN = μN γ . Then γ = lim γ , and so it is enough to prove that γ is surjective for every N. ←− N N Thus from now on we may assume that the Galois covering ξ has finite associated group. Let H = f(G(Γ|)) and let t1,...,tn be representatives of the left cosets of H in G(Σ|). Observe that Hγ(Γ) = γ(Γ), since f(g)γ(m) = γ(gm) (g ∈ G(Γ |), m ∈ Γ);so n Σ = tiγ(Γ). i=1

We claim that the subgraphs tiγ(Γ) (i = 1,...,n) of Σ are clopen. They are obviously closed. Hence it sufﬁces to show that they are disjoint. To see this, let s ∈ tiγ(Γ) ∩ tj γ(Γ),forsomei, j ∈{1,...,n}. Then tiγ(m1) = tj γ(m2),for some m1,m2 ∈ Γ . Hence ζ(m1) = ζ(m2), and therefore m1 = gm2,forsome g ∈ G(Γ |). Then tif(g)γ(m2) = tj γ(m2). Since G(Σ|) acts freely on Σ, one concludes that tif(g)= tj ; hence, i = j. This proves the claim. Since Σ is connected, it follows (see Lemma 2.1.9) that n = 1, i.e., γ(Γ)= Σ. (b) We just need to prove that γ coincides with the natural epimorphism Γ −→ Ker(f )\Γ.

If gm1 = m2,form1,m2 ∈ Γ , g ∈ Ker(f ), then

γ(m2) = γ(gm1) = f(g)γ(m1) = γ(m1).

Conversely, assume that γ(m1) = γ(m2),forsomem1,m2 ∈ Γ . Then there exists a g ∈ G(Γ |) such that m1 = gm2. Therefore,

γ(m2) = γ(m1) = f(g)γ(m2); so f(g)= 1, since G(Σ|) acts freely on Σ, i.e., g ∈ Ker(f ). 3.1 Galois Coverings 71

Corollary 3.1.6 Let

γ Γ1 Γ2

ζ1 ζ2 δ 1 2 be a commutative diagram of proﬁnite graphs and qmorphisms such that ζ1 and ζ2 are connected Galois coverings. Then γ is surjective if and only if δ is surjective.

Proof If γ is a surjection, it is clear that so is δ. Conversely, assume that δ is a surjection. By Proposition 3.1.4, there exists a continuous homomorphism

f : G(Γ1|1) −→ G(Γ2|2) which is compatible with γ . It follows that the restriction

ζ2|Im(γ ) : Im(γ ) −→ 2 of ζ2 to Im(γ ) is a Galois covering with associated group G(Im(γ )|2) = Im(f ). Then, by Proposition 3.1.5, the inclusion Im(γ ) → Γ2 is surjective; thus Im(γ ) = Γ2, i.e., γ is surjective.

Lemma 3.1.7 Let ζ : Γ → be a connected Galois covering and let Σ be a connected proﬁnite graph. If β1,β2 : Σ → Γ are morphisms of proﬁnite graphs with ζβ1 = ζβ2, and β1(m) = β2(m), for some m ∈ Σ, then β1 = β2.

Proof Assume ﬁrst that Γ is ﬁnite. Let Σ be the set of elements of Σ where β1 and

β2 coincide. Then Σ is a closed nonempty subgraph of Σ. Since Γ is ﬁnite and = −1 ∩ −1 Σ β1 (x) β2 (x) , x∈Γ it follows that Σ is clopen in Σ. Suppose Σ = Σ . Then there exists an edge e of

Σ with e/∈ E(Σ ) such that dj (e) ∈ Σ for j = 0 or 1, because otherwise Σ − Σ is an open subgraph of Σ and Σ = Σ ∪ (Σ − Σ ) would be a disconnected graph

(see Lemma 2.1.9), contradicting our hypothesis. Say d0(e) ∈ Σ . Hence

d0 β1(e) = β1 d0(e) = β2 d0(e) = d0 β2(e) .

Since ζ(β1(e)) = ζ(β2(e)), there exists a nontrivial element g ∈ G(Γ |) such that gβ1(e) = β2(e). Therefore

gd0 β1(e) = d0 β2(e) = d0 β1(e) , contradicting the freeness of the action of G(Γ |) on Γ . 72 3 The Fundamental Group of a Proﬁnite Graph

Suppose now that Γ is inﬁnite. Then, by Proposition 3.1.3, there exists a decomposition ζ = lim ζ of ζ as an inverse limit of ﬁnite Galois coverings ζ : Γ → . ←− i i i i Let ϕi be the projection of Γ onto Γi . By the case above, ϕiβ1 = ϕiβ2. Since this is valid for every i, we deduce that β1 = β2, as required.

3.2 G(Γ |) as a Subgroup of Aut(Γ )

Let ζ : Γ → be a Galois covering of a proﬁnite graph , and let G = G(Γ |) be the associated proﬁnite group. Each element g of G determines a continuous automorphism

νg : Γ −→ Γ given by νg(m) = gm (g ∈ G, m ∈ Γ). Moreover, the map

ν : G −→ Aut(Γ ) which sends g to νg is a homomorphism, and it is injective because G acts freely on Γ . The ﬁrst result in this section says that we may think of the topological group G as a closed subgroup of Aut(Γ ) if we identify G with ν(G) and assume that Aut(Γ ) is endowed with the compact-open topology. Recall that the compact-open topology on Aut(Γ ) is generated by a sub-base of open subsets of the form B(K,U) = f ∈ Aut(Γ ) f(K)⊆ U , where K ranges over the compact subsets of Γ and U ranges over the open subsets of Γ (cf. Bourbaki 1989,X,3,4).

Proposition 3.2.1 Let ζ : Γ → be a Galois covering of a profinite graph , and let G = G(Γ |) be the associated profinite group. Consider the group Aut(Γ ) of automorphisms of the profinite graph endowed with the compact-open topology. Then G is naturally embedded in Aut(Γ ) as a topological subgroup, i.e., there exists a topological isomorphism of G with a closed subgroup of Aut(Γ ).

Proof We claim first that the monomorphism ν : G → Aut(Γ ) defined above is continuous. To see this, let K be a compact subset of Γ and let U be an open subset of Γ .Let B(K,U) = f ∈ Aut(Γ ) f(K)⊆ U be the corresponding sub-basic open subset of Aut(Γ ). It suffices to show that ν−1(B(K, U)) is open in G. To see this it is enough to prove that if g ∈ G and νg ∈ B(K,U), then there exists an open neighbourhood W of g in G contained in ν−1(B(K, U)). We shall check that this is indeed the case. Since gK ⊆ U, for each 3.2 G(Γ |) as a Subgroup of Aut(Γ ) 73 x ∈ K, there exist open neighbourhoods Wx of g in G and Vx of x in Γ such that ⊆ WxVx U. Since K is compact, there exist finitely many points x1,...,xn in K so n = that i=1 Vxi K. Then n = W Wxi i=1 is the desired neighbourhood of g. This proves the claim. Finally, since G is compact, the above implies that ν maps G homeomorphically onto Im(ν), if we can prove the following assertion: the compact-open topology on Aut(Γ ) is Hausdorff. To show this, let f1,f2 ∈ Aut(Γ ) be distinct. Then there exists an m ∈ Γ with f1(m) = f2(m).LetUi be a neighbourhood of fi(m) (i = 1, 2) such that U1 ∩ U2 =∅. Then clearly fi ∈ B({m},Ui)(i= 1, 2) and B({m},U1) ∩ B({m},U2) =∅, proving the assertion.

Let ζ : Γ → be a connected Galois covering of a proﬁnite graph , and put G = G(Γ |). By the above proposition we may think of G as a subgroup of Aut(Γ ).Let

H = NAut(Γ )(G), the normalizer of G in Aut(Γ ). Then H is a closed subgroup of Aut(Γ ). Observe that if f ∈ H, then f induces a map

Φ(f) : = G\Γ −→ = G\Γ deﬁned by Φ(f )(Gm) = Gf (m) (m ∈ Γ).

Proposition 3.2.2 (a) If f ∈ H, then Φ(f) is a continuous automorphism of the profinite graph , i.e., Φ(f) ∈ Aut(). (b) The map Φ : H −→ Aut() is a continuous homomorphism. (c) Ker(Φ) = G. (d) If is finite, then the homomorphism Φ : H → Aut() is open and the group H = NAut(Γ )(G) is profinite.

Proof (a) It is easy to verify that Φ(f) is a map of graphs and a bijection. We shall prove that it is continuous. Consider an open subset U of . Then

Φ(f)−1(U) = ζf −1 ζ −1(U) , and this is an open set since f is continuous and ζ is continuous and open. 74 3 The Fundamental Group of a Proﬁnite Graph

(b) Consider a sub-basic open subset B(K,U) of Aut(), where K is a compact subset of and U is an open subset of . Then putting K˜ = ζ −1(K) and U˜ = ζ −1(U),wehave

Φ(f)−1 B(K,U) = B(K,˜ U),˜ which is open in Aut(Γ ). (c) Clearly G ≤ Ker(Φ). Conversely, let f ∈ H be such that Φ(f) = Id. Choose m ∈ Γ . Then f(m)= gm,forsomeg ∈ G. Deﬁne

g˜ : Γ −→ Γ to be the graph morphism which is multiplication by g. Then ζf = ζ g˜, since f ∈ Ker(Φ). Therefore f =˜g by Lemma 3.1.7, since f(m)=˜g(m). (d) The map Φ is open since Aut() is discrete. If follows that H/G =∼ Im(Φ), as topological groups. Since H/G is finite and Ker(Φ) = G is profinite, the group H is profinite.

3.3 Universal Galois Coverings and Fundamental Groups

Let C be a pseudovariety of ﬁnite groups. A Galois covering ζ : Γ → is said to be a Galois C-covering if its associated group G(Γ |) is a pro-C group. Note that since a Galois C-covering is a Galois covering, all the results of the previous sections are valid for Galois C-coverings. A connected Galois C-covering

ζ : Γ˜ −→ Γ of a connected profinite graph Γ is said to be universal if the following universal property holds: given any qmorphism β : Γ → to a connected profinite graph , any connected Galois C-covering ξ : Σ → , and any points m ∈ Γ˜ , s ∈ Σ such that βζ(m) = ξ(s), there exists a qmorphism of profinite graphs α : Γ˜ → Σ, such that βζ = ξα and α(m) = s

α Γ˜ Σ

ζ ξ β Γ

We say that α lifts β, or that α is a lifting (qmorphism) of β. Once m ∈ Γ˜ and s ∈ Σ with βζ(m) = ξ(s) are given, the lifting qmorphism α is unique (see Lemma 3.1.7). Note also that if the map β is surjective, so is α by Corollary 3.1.6. Furthermore, it follows from Proposition 3.1.3 that it is sufﬁcient to check the universal property above for ﬁnite Galois C-coverings ξ : Σ → . 3.3 Universal Galois Coverings and Fundamental Groups 75

We shall prove the existence of universal Galois coverings in Proposition 3.5.3 and Proposition 3.5.6 below. We prove uniqueness in the following proposition.

Proposition 3.3.1 Let ζ : Γ˜ → Γ be a universal Galois C-covering of a profinite connected graph Γ . (a) Assume that α : Γ˜ → Γ˜ is a morphism of profinite graphs such that ζα = ζ . Then α is an automorphism. (b) A universal Galois C-covering is unique, if it exists. More precisely, if ζ : Γ˜ → Γ is another universal Galois C-covering of Γ , then there exists an isomorphism α : Γ˜ → Γ˜ of profinite graphs such that ζ α = ζ .

Proof (a) Let m ∈ Γ˜ and set m = α(m). By the universality of ζ , there exists a morphism β : Γ˜ → Γ˜ such that β(m ) = m and ζβ = ζ . Since βα(m) = m and = = = αβ(m ) m ,wehavebyLemma3.1.7 that βα αβ idΓ˜ . Thus α is an automorphism. (b) Choose m ∈ Γ˜ , m ∈ Γ˜ such that ζ(m) = ζ (m ). Then there exist morphisms α : Γ˜ → Γ˜ , α : Γ˜ → Γ˜ with α(m) = m , α (m ) = m, which are lift- = ing morphisms of the identity morphism of Γ . By the argument in (a), α α idΓ˜ , = αα idΓ˜ . Thus α is an isomorphism.

C = ˜ | C The profinite group π1 (Γ ) G(Γ Γ)is called the fundamental pro- group of the connected profinite graph Γ . By Proposition 3.3.1(b) and Proposition 3.1.5(b), the fundamental pro-C group of Γ is well-defined up to isomorphism. We say that C C = a connected profinite graph Γ is -simply connected if π1 (Γ ) 1. In Proposi- tion 3.3.3 and in Sect. 3.10 we shall study the relationship between simple connectivity and being a C-tree. Next we show that forming universal Galois C-coverings commutes with inverse limits.

Proposition 3.3.2 Let {Γi,ϕij ,I} be an inverse system of proﬁnite connected graphs Γi over a poset (I, ), and let = Γ lim←− Γi. i∈I ˜ For each i ∈ I , let ζi : Γi → Γi be a universal Galois C-covering of Γi . Then

(a) The Galois C-coverings ζi form an inverse system over I and = ζ lim←− ζi i∈I

is a universal Galois C-covering ζ : Γ˜ → Γ of Γ . C (b) The fundamental groups π1 (Γi) form an inverse system over I and C = C π1 (Γ ) lim←− π1 (Γi). i∈I 76 3 The Fundamental Group of a Proﬁnite Graph

Proof Denote by ϕi : Γ → Γi (i ∈ I)the projection. Fix some m ∈ Γ and put mi = ˜ ϕi(m) (i ∈ I). For each i ∈ I , choose m˜ i ∈ Γi such that ζi(m˜ i) = mi For i j in I , ˜ ˜ let ϕ˜ij : Γi → Γj be the (unique) lifting of ϕij : Γi → Γj such that ϕ˜ij (m˜ i) =˜mj . ˜ Then {Γi, ϕ˜ij ,I} is an inverse system over I , and therefore {ζi,(ϕ˜ij ,ϕij ), I} is an inverse system over I of Galois coverings. Now, (ϕ˜ij ,ϕij ) induces a canonically : C → C ˜ deﬁned continuous homomorphism fij π1 (Γi) π1 (Γj ) compatible with ϕij { C } (see Proposition 3.1.4). Therefore, π1 (Γi), fij ,I is an inverse system over I . Put = C = ˜ G lim←− π1 (Γi) and Γ lim←− Γi. i∈I i∈I

Clearly G acts freely on Γ and G\Γ = Γ . So, the quotient map ζ : Γ → Γ is a Galois C-covering. Moreover, ζ can be identified with the map induced by the maps ζ , i.e., ζ = lim ∈ ζ . i ←− i I i We claim that ζ is a universal Galois C-covering of Γ (this will show that Γ = Γ˜ and ζ = ζ , i.e., that part (a) holds). Let ξ : Σ → be a connected finite Galois C-covering and let μ : Γ → be a morphism of profinite graphs. Choose points m ∈ Γ and s ∈ Σ with μζ (m ) = ξ(s). Since is finite, there exists some ∈ : → = i0 I and a morphism μ0 Γi0 with μ0ϕi0 μ (see Lemma 2.1.5).

˜ Γ Γi Σ α 0 i0 μ˜ 0

ζ ζi ξ

ϕi0 μ0 Γ Γi0

˜ : ˜ → ˜ = : Let μ0 Γi0 Σ be the lifting morphism of μ0 with μ0αi0 (m ) s, where αi0 → ˜ ˜ Γ Γi0 is the projection. Then obviously, μ0αi0 is the required lifting morphism of μ. This proves the claim and part (a). Part (b) follows from Proposition 3.1.2.

Proposition 3.3.3 (a) Let T be a finite tree. Then T is C-simply connected for every pseudovariety of finite groups C. (b) Let Γ be a profinite graph which is an inverse limit of finite trees. Then, for every pseudovariety of finite groups C, Γ is both a C-tree and a C-simply connected profinite graph.

Proof To prove part (a) we need to show that the identity morphism T → T is a universal Galois C-covering. Let β : T → be a qmorphism of T to a ﬁnite graph and let ζ : Σ → be a ﬁnite Galois C-covering. Fix m ∈ T and s ∈ Σ with β(m) = ζ(s).LetT be a subtree of T containing m and maximal with respect to the property that there exists a qmorphism α : T → Σ such that ζα = β|T and 3.4 0-Transversals and 0-Sections 77

α(m) = s. We need to prove that T = T . Suppose not. Then there exists an edge e ∈ E(T ) − E(T ) with one of its vertices in T ,sayd1(e). Since ζ is a Galois −1 covering, d (α(d1(e))) contains an edge e with ζ(e ) = β(e). Note that d0(e) = 1 d1(e) since T is a tree. Extend α to a morphism of graphs on T ∪{e}∪{d0(e)} by putting α(e) = e and α(d0(e)) = d0(e ). This contradicts the maximality of T , and so T = T . (Note that this argument is independent of the class C.) (b) By Proposition 2.4.3(d), Γ is a C-tree. That it is also C-simply connected follows from Proposition 3.3.2(b).

For a converse of Proposition 3.3.3(a), see Corollary 3.5.5.

Exercise 3.3.4 Let C be a pseudovariety of ﬁnite groups. (a) Let L(0) be the loop 0 • 0 Prove that the universal Galois C-covering of L(0) is the Cayley graph Γ(ZCˆ, {1}) of the free pro-C group ZCˆ of rank 1 (written additively) with respect { } C =∼ to its generating subset 1 . Moreover, π1 (L(0)) ZCˆ. (b) Let n be a natural number. Consider the ﬁnite graph

0 12n − 1 n L(n) : •••··· •• − 0 1 n 1 n Prove that the universal Galois C-covering graph of L(n) consists of the Cayley graph Γ(ZCˆ, {1}) of the free pro-C group ZCˆ of rank 1 with respect to the subset {1} to which one attaches a copy of the ﬁnite tree 0 12n − 1 n I(n): •••··· •• 0 1 n − 1

{ } C =∼ C =∼ to each point of Γ(ZCˆ, 1 ). Moreover, π1 (L(n)) π1 (L(0)) ZCˆ. (Hint: see Proposition 3.5.3 below.)

3.4 0-Transversals and 0-Sections

Let Γ be a connected profinite graph. A spanning profinite subgraph of Γ is a profinite subgraph T of Γ with V(T)= V(Γ).IfΓ is finite, then it has a spanning subtree; more generally, it is well-known that any connected abstract graph admits a spanning subtree (cf. Serre 1980, Proposition I.11). However, in general it need not be the case that a connected profinite graph has a spanning profinite subgraph which is a C-tree or a C-simply connected profinite subgraph, as the following example shows. 78 3 The Fundamental Group of a Profinite Graph

Example 3.4.1 (Connected graph with no spanning C-simply connected profinite subgraph or C-tree) Let N ={0, 1, 2,...} be the set of natural numbers with the discrete topology and let N¯ = N ∪. {∞} be the one-point compactification of N.De- fine a profinite graph L = N¯ ×{0, 1} with space of vertices and edges V(L)= i = (i, 0) i ∈ N¯ ,E(L)= i = (i, 1) i ∈ N¯ , and incidence maps d0,d1 : E(L) → V(L)defined by d0(i) = i and d1(i) = i + 1 (i ∈ N¯ ), where ∞+1 =∞:

0 123 ∞ L : ••••··· • 0 1 2 ∞

Observe that V(L) and E(L) are disjoint and they are both proﬁnite spaces. Note that L is the inverse limit of the following ﬁnite connected graphs L(n) (n ≥ 0)

0 12n − 1 ∞ L(n) : •••··· •• − 0 1 n 1 ∞ where the canonical map L(n + 1) → L(n) sends i to i identically, if i ≤ n − 1, and it sends n and ∞ to ∞. One deduces that L is a connected profinite graph. We claim that any connected profinite subgraph L of L which contains all the vertices of L coincides with L; indeed, since L is connected and contains all the vertices of L, it must contain all the edges of the form i (i = 0, 1,...); therefore, since L is compact, it also contains ∞; this proves the claim. On the other hand, if C is a C =∼ C =∼ pseudovariety of finite groups, one easily sees that π1 (L) π1 (L(0)) ZCˆ (see Exercise 3.3.4 or Proposition 3.5.3 below). Hence L does not contain any spanning C-simply connected profinite subgraph. Similarly, L does not contain any spanning C-subtree since obviously L is not a C-tree (see Exercise 2.4.4).

Let G be a profinite group that acts on a profinite graph , and let ϕ : → Γ = G\ be the canonical quotient map. Let Λ be a profinite subgraph of Γ ;if there is a profinite subgraph Λ of such that ϕ|Λ is a monomorphism of graphs with ϕ|Λ (Λ ) = Λ, we say that Λ is a lifting of Λ. In contrast with the situation for abstract groups that act on abstract graphs (cf. Serre 1980, Proposition I.14), a general π-subtree or C-simply connected profinite subgraph of Γ need not have a lifting to an isomorphic profinite subgraph of , as the following example shows.

Example 3.4.2 (Quotient G-graph with no lifting of trees or simply connected subgraphs) Let X be a proﬁnite space on which a pro-π group G acts continuously in such a way that the canonical epimorphism ϕ : X → G\X does not admit a continuous section (see Sect. 1.3). Construct a proﬁnite graph C = C(X,P), where P is a point not in X, as follows: C = V(C)∪. E(C), where V(C)= X ∪. {P },E(C)= (x, P ) x ∈ X ,d0(x, P ) = P, d1(x, P ) = x. 3.4 0-Transversals and 0-Sections 79

We term C the cone of X. Extend the action of G on X to an action of G on V(C) by letting every g ∈ G ﬁx P . Deﬁne an action of G on E(C) as follows:

g(x,P) = (gx, P ), (g ∈ G, x ∈ X).

One checks that this defines a continuous action of G on the profinite graph C. We claim that C is both a C-tree and a C-simply connected graph, for any pseudovariety of finite groups C; indeed, write X as an inverse limit of finite quotient spaces Xi ; then C = lim C(X ,P); ←− i therefore C is the inverse limit of finite trees, and so the claim follows from Propo- sition 3.3.3(b). The quotient graph of C under the action of G is the cone of G\X:

G\C = C(G\X, P ).

In particular, G\C is a C-tree and a C-simply connected graph which does not have a lifting to a proﬁnite subgraph of C(X,P), i.e., there is no morphism of proﬁnite graphs ψ : C(G\X, P ) → C(X,P) such that ϕψ = idC(G\X,P ).

Let G be a proﬁnite group that acts on a connected proﬁnite graph , and let ϕ : → Γ = G\ be the canonical quotient map. A G-transversal or a transversal of ϕ is a closed subset J of such that ϕ|J : J → Γ is a homeomorphism. Associated with such a transversal there is a continuous G-section or section of ϕ

j : Γ −→ , i.e., a continuous mapping such that ϕj = idΓ and j(Γ)= J . Note that, in general, J is not a graph. We say that a transversal J is a 0-transversal if d0(m) ∈ J , for each m ∈ J ; in that case we refer to j as a 0-section. Note that if j is a 0-section, then

jd0 = d0j.

If it happens that the quotient graph Γ = G\ admits a spanning C-simply connected proﬁnite subgraph, then we say that a 0-transversal J is a fundamental 0- transversal and the corresponding 0-section j : Γ → is a fundamental 0-section if for some spanning C-simply connected proﬁnite subgraph T of Γ , T = j(T) is a lifting of T , i.e., the restriction of j to T is a morphism of graphs. Example 3.4.2 above shows that G-transversals do not exist in general. However, the following result proves that their existence is guaranteed when the action of G is free. See also Proposition 3.4.5, Corollary 3.7.2 and Theorem 3.7.4.

Lemma 3.4.3 Let ζ : → Γ be a Galois C-covering of a profinite graph Γ with associated group G = G(|Γ). Assume that Λ is a lifting of a profinite subgraph Λ of Γ . Then there exists a 0-transversal J ⊆ of ζ such that Λ ⊆ J . 80 3 The Fundamental Group of a Profinite Graph

Proof Note that G\V()= V(Γ). Since V()is a proﬁnite space and G acts on it freely, there is a continuous section

jV : V(Γ)−→ V() of ζ|V() such that {d0(m) | m ∈ Λ }⊆jV (V (Γ )) (see Sect. 1.3). Deﬁne

= −1 J d0 jV V(Γ) .

Note that Λ ⊆ J . Clearly d0(J ) ⊆ J . It remains to prove that the restriction ζ|J : J → Γ of ζ to J is a homeomorphism of topological spaces. Since J is compact, it sufﬁces to prove that ζ|J is a bijection. Let m ∈ Γ , and let m˜ ∈ be such that ζ(m)˜ = m. Put v = jV d0(m). Since ζd0(m)˜ = ζ(v)= d0(m), there exists some g ∈ G with gd0(m)˜ = v. Put m = gm˜ . Then ζ(m ) = m, and d0(m ) = v, i.e., m ∈ J .Soζ|J is onto. Now, if m1,m2 ∈ J and ζ(m1) = ζ(m2), then there exists some g ∈ G with gm1 = m2, and so gd0(m1) = d0(m2). Since d0(m1), d0(m2) ∈ jV (V (Γ )), we deduce that d0(m1) = d0(m2); and since the action of G is free, g = 1. Therefore m1 = m2, proving that ζ|J is also an injection.

Let ζ : Γ˜ → Γ be the universal Galois C-covering of a profinite graph Γ .By Lemma 3.4.3 there exists a continuous 0-section j : Γ → Γ˜ of ζ .LetJ = j(Γ) be the corresponding 0-transversal. Associated with this transversal we are going to define two continuous maps = : ˜ −→ C = : −→ C κ κj Γ π1 (Γ ) and χ χj Γ π1 (Γ ) (3.1) which will play an important role later. If m˜ ∈ Γ˜ , define κ(m)˜ to be the unique C element of π1 (Γ ) such that

κ(m)˜ jζ(m)˜ =˜m.

For an arbitrary m ∈ Γ one has

ζd1j(m)= d1ζj(m) = d1(m) = ζjd1(m). C We deﬁne χ(m) to be the unique element of π1 (Γ ) such that

χ(m) jd1(m) = d1j(m).

Lemma 3.4.4 The following properties hold for the functions κ = κj and χ = χj deﬁned above.

(a) κd1j(m)= χ(m), for all m ∈ Γ . ˜ = ˜ ˜ ∈ ˜ ∈ C (b) κ(hm) hκ(m), for all m Γ,h π1 (Γ ). ˜ (c) κ(m)˜ = κd0(m)˜ , for all m˜ ∈ Γ . (d) κj (m) = 1, for all m ∈ Γ ; χ(v)= 1, for all v ∈ V(Γ). ˜ (e) κ(m)(χζ(˜ m))˜ = κd1(m)˜ , for all m˜ ∈ Γ . (f) The maps κ and χ are continuous. 3.4 0-Transversals and 0-Sections 81

Proof Parts (a), (b), (c) and (d) follow easily from the deﬁnitions of κ and χ.To show (e) note that

χζ(m)˜ jζd1(m)˜ = χζ(m)˜ jd1ζ(m)˜ = d1jζ(m)˜

−1 −1 = d1 κ(m)˜ m˜ = κ(m)˜ d1(m),˜ so that κ(m)(χζ(˜ m))(jζ˜ d1(m))˜ = d1(m)˜ , proving (e). In view of (a), to prove part (f) it sufﬁces to show that κ is continuous since d1 and j are continuous. Showing the continuity of κ is equivalent to proving that the graph = ˜ ˜ ˜ ∈ ˜ ⊆ ˜ × C Gr(κ) m, κ(m) m Γ Γ π1 (Γ ) of the function κ is closed: indeed, just observe that then Gr(κ) would be compact and so the natural projection Gr(κ) → Γ˜ , which is bijective, would be a homeomorphism; since κ is the inverse map of this homeomorphism composed with the → C projection Gr(κ) π1 (Γ ), the result would follow. To see that Gr(κ) is closed, deﬁne a map : ˜ × C −→ ˜ γ Γ π1 (Γ ) Γ by

˜ = −1 ˜ ∈ C ˜ ∈ ˜ γ(m, h) h m h π1 (Γ ), m Γ . Clearly γ is continuous, so it will sufﬁce to show that Gr(κ) = γ −1(J ). Note that if m˜ ∈ Γ˜ , then jζ(m)˜ = κ(m)˜ −1m˜ ; therefore

m,˜ κ(m)˜ ∈ γ −1 jζ(m)˜ ∈ γ −1(J ).

Conversely, if (m,˜ h) ∈ γ −1(J ), then h−1m˜ = j(m),forsomem ∈ Γ ; so, using (d) and (b), κ(m)˜ = κ(hj(m)) = h(κj (m)) = h, i.e., (m,˜ h) = (m,˜ κ(m))˜ ∈ Gr(κ),as needed.

One can sharpen Lemma 3.4.3 when the quotient graph Γ = G\ is ﬁnite.

Proposition 3.4.5 Let a profinite group G act on a connected profinite graph and let ϕ : → Γ = G\ be the corresponding projection. (a) Let T be a finite subtree of the graph Γ and let T be a finite subtree of that ϕ sends injectively into T . Then T lifts to a subtree of containing T . (b) Assume further that the action of G on is free and that the quotient graph Γ = G\ is finite. Let m0 ∈ . Then there exists a fundamental 0-transversal J in containing m0.

Proof (a) Let L be the set of finite subtrees of containing T which are sent L injectively into T by means of ϕ.LetT0 be a maximal element of with respect to inclusion, and let T0 be its image in T . Suppose that T0 = T . Since T is finite 82 3 The Fundamental Group of a Profinite Graph and connected, there exists an edge e of T not belonging to T0 such that one of the vertices of e is in T0,sayd0(e) ∈ V(T0); then d1(e)∈ / V(T0).Letv be a vertex of

T whose image in T is d0(e).Lete ∈ with ϕ(e ) = e. Since v and d0(e ) are in 0 the same G-orbit, there exists some g ∈ G with gd0(e ) = v . Define e = ge . Then = = ∪{ }∈L d0(e ) v and ϕ(e ) e. Since T0 e ,d1(e ) , this contradict the maximality = = of T0. Thus ϕ(T0) T0 T , as desired. (b) Since Γ is finite and connected, it has a subtree T with V(T)= V(Γ)(i.e., T is a spanning simply connected profinite subgraph of Γ ). By part (a) there exists −1 a lifting T of T such that d0(m0) ∈ V(T ). Define J = d (V (T )). Note that 0 m0 ∈ J and T ⊆ J . Then arguing as in the proof of Lemma 3.4.3, we see that J is a 0-transversal, and since T is a lifting of a maximal tree of Γ , J is a fundamental 0-transversal. Equivalently, one can describe J more explicitly: for each edge e ∈

Γ − T , choose e ∈ such that d0(e ) ∈ T and ϕ(e ) = e (this can be done since d0(e) ∈ T , and every vertex of is in the G-orbit of a vertex of T ; furthermore, such e is unique because G acts freely on ); then J consists of T together with all the chosen edges e (e ∈ Γ − T ).

3.5 Existence of Universal Coverings

In this section we prove the existence of universal Galois C-coverings of a connected profinite graph Γ . We begin with a general construction. When a profinite graph admits a C-simply connected profinite subgraph, this provides an explicit construction of a universal Galois C-covering. This is the case, in particular, for all finite graphs.

3.5.1 A Special Construction of Galois Coverings Let Γ be a connected proﬁnite graph and let T be a connected proﬁnite subgraph of Γ with V(T)= V(Γ)(T need not equal Γ ). Denote by X = Γ/T the corresponding quotient space with canonical map ω : Γ −→ X = Γ/T.

Consider the element ∗=ω(T) as a distinguished point of X.LetF = FC(X, ∗) be the free pro-C group on the pointed profinite space (X, ∗) and think of (X, ∗) as being a subspace of FC(X, ∗) in the natural way. Define a profinite graph ΥC(Γ, T ) as follows:

ΥC(Γ, T ) = FC(X, ∗) × Γ, V ΥC(Γ, T ) = FC(X, ∗) × V(Γ), d0(r, m) = (r, d0(m)) and d1(r, m) = (rω(m), d1(m)) (r ∈ F,m∈ Γ).

Next deﬁne an action of F on the graph ΥC(Γ, T ) by

r (r, m) = r r, m r, r ∈ F,m∈ Γ . 3.5 Existence of Universal Coverings 83

Clearly this is a free action and F \ΥC(Γ, T ) = Γ . Therefore the natural epimorphism

υ : ΥC(Γ, T ) −→ Γ that sends (r, m) to m (r ∈ F,m∈ Γ ) is a Galois C-covering.

Lemma 3.5.2 The Galois covering υ : ΥC(Γ, T ) → Γ is connected.

Proof We need to show that the graph Υ = ΥC(Γ, T ) is connected. Write Γ as an inverse limit = Γ lim←− Γi i∈I of ﬁnite quotient graphs (see Proposition 2.1.4), and let ρi : Γ → Γi be the canonical epimorphism (i ∈ I). Put Ti = ρi(T ). Then V(Ti) = V(Γi) and Ti is connected. We view Xi = Γi/Ti as a pointed space whose distinguished point ∗ is the image of Ti in Xi . One has ∗ = = ∗ (X, ) Γ/T lim←− (Xi, ), i∈I so that (RZ, Proposition 3.3.9) ∗ = ∗ F(X, ) lim←− F(Xi, ). i∈I Clearly = Υ(Γ,T) lim←− Υ(Γi,Ti). i∈I Hence we may assume that Γ is ﬁnite. If U is an open normal subgroup of F(X,∗), consider the continuous map

w wU : Γ −→ (X, ∗)→ F(X,∗) −→ F(X,∗)/U. Deﬁne the structure of a graph on

ΥU = F(X,∗)/U × Γ by imitating the construction above: its vertices are

V(ΥU ) = F(X,∗)/U × V(Γ), while d0(τ, m) = τ,d0(m) ,d1(τ, m) = τwU (m), d1(m) τ ∈ F(X,∗)/U, m ∈ Γ .

Then ΥU is a ﬁnite graph on which F(X,∗)/U acts, and = Υ(Γ,T) lim←− ΥU . U 84 3 The Fundamental Group of a Proﬁnite Graph

Therefore it sufﬁces to show that each ΥU is connected. To verify this consider the connected subgraph of ΥU deﬁned as

= {1U}×Γ ∪ d1 {1U}×Γ , and observe that ΥU = (F (X, ∗)/U). Next note that wU (Γ ) generates the group F(X,∗)/U; moreover, the graphs and wU (m) have the vertex d1({1U},m)= (wU (m), m) in common (m ∈ Γ). It follows from Lemma 2.2.4(a) that ΥU is connected.

We aim to show that

υ : ΥC(Γ, T ) → Γ is universal if T is a spanning C-simply connected proﬁnite subgraph of Γ .We prove this for a ﬁnite graph Γ in the next proposition and in Theorem 3.7.4 in full generality.

Proposition 3.5.3 Let Γ be a ﬁnite connected graph and let T be a maximal subtree of Γ (T is a spanning C-simply connected proﬁnite subgraph of Γ according to Proposition 3.3.3(a)). Then one has the following properties.

(a) The Galois C-covering υ : ΥC(Γ, T ) → Γ constructed in 3.5.1 is universal. (b) Let (X, ∗) = (Γ /T , ∗); then C = ∗ π1 (Γ ) FC(X, ) is a free pro-C group of ﬁnite rank |Γ |−|T |. (c) The universal Galois C-covering υ : ΥC(Γ, T ) → Γ is independent of the maximal subtree T chosen.

Proof Part (b) follows from part (a) and the construction of ΥC(Γ, T ). Part (c) is a consequence of (a) and the uniqueness of universal coverings. To prove (a) we need to check the appropriate universal property for υ : ΥC(Γ, T ) → Γ .Letβ : Γ → be a qmorphism into a connected ﬁnite graph and let ξ : Σ → be a ﬁnite connected Galois C-covering. Consider the pullback of β and ξ

α Γ Σ

ζ ξ

Γ β

Then Γ is the proﬁnite graph Γ = (m, s) ∈ Γ × Σ β(m) = ξ(s) 3.5 Existence of Universal Coverings 85 whose space of vertices is

V Γ = Γ ∩ V(Γ)× V(Σ) , and with incidence maps deﬁned by

di(m, s) = di(m), di(s) i = 0, 1; (m, s) ∈ Γ .

The morphisms α and ζ are the obvious projections. Deﬁne an action of G(Σ|) on Γ by

g(m,s) = (m, gs) g ∈ G(Σ|), (m, s) ∈ Γ . This action is free and the quotient graph G(Σ|)\Γ = (m, k) ∈ Γ × β(m) = k can be identiﬁed with Γ , so that the projection ζ : Γ → Γ coincides with the natural epimorphism Γ → G(Σ|)\Γ . Let u0 = (r, z) ∈ ΥC(Γ, T ) = FC(X, ∗) × Γ and s0 ∈ Σ be such that βυ(u0) = ξ(s0). We need to prove that there exists a qmorphism

α : ΥC(Γ, T ) −→ Σ of proﬁnite graphs such that ξα = βυ and α(u0) = s0. Since the map ΥC(Γ, T ) → − ΥC(Γ, T ) given by u → r 1u(u∈ ΥC(Γ, T )) is an isomorphism of proﬁnite graphs that commutes with υ, we may assume that u0 = (1,z). Let ∗ ∗ Γ = Γ (z, s0) be the connected component of (z, s0) in Γ .

ΥC(Γ, T )

α θ

Γ ∗

α Γ Σ υ ∗ j ζ

ζ ξ

Γ β 86 3 The Fundamental Group of a Proﬁnite Graph

We claim that ∗ = : ∗ −→ ζ ζ|Γ ∗ Γ Γ is a Galois C-covering of Γ . Indeed, let G∗ be the maximal subgroup of G(Σ|) ∗ ∗ ∗ ∗ that leaves Γ invariant. Suppose m1,m2 ∈ Γ . Then ζ (m1) = ζ (m2) if and only ∗ ∗ if gm1 = m2,forsomeg ∈ G(Σ|); if this is the case, m2 ∈ Γ ∩ gΓ , hence gΓ ∗ ∪ Γ ∗ is connected (see Lemma 2.1.7), and therefore

gΓ ∗ ∪ Γ ∗ = Γ ∗.

It follows that g ∈ G∗. Thus ζ ∗ coincides with the natural epimorphism Γ ∗ → G∗\Γ ∗. To prove the claim it remains to show that ζ ∗(Γ ∗) = Γ . Suppose not. Then, since Γ is a ﬁnite and connected graph, there exists an edge e ∈ E(Γ ) − ζ ∗(Γ ∗) ∈ ∗ ∗ = = = such that di0 (e) V(ζ (Γ )),fori0 0ori0 1. Put w di0 (e) and choose w∗ ∈ Γ ∗ with ζ ∗(w∗) = w.Lete ∈ Γ with ζ (e ) = e. Then w∗ is a vertex of he , ∗ for some h ∈ G(Σ|). Hence the graph Γ ∪{he ,d0(he ), d1(he )} is connected and therefore it is contained in Γ ∗. It follows that

e = ζ he ∈ ζ ∗ Γ ∗ , a contradiction. Thus ζ ∗(Γ ∗) = Γ , i.e., ζ ∗ is a Galois C-covering of Γ , proving the claim. Since Γ is ﬁnite, there exists a fundamental G∗-transversal J in Γ ∗ lifting T such that (z, s0) ∈ J (see Proposition 3.4.5(b)). In particular, there is a bijection

j : Γ −→ J

∗ such that ζ j = idΓ and j|T is a graph morphism. Note that j(z) = (z, s0).Let x ∈ X −{∗}=Γ − T ; then d1(j (x)) and j(d1(x)) have the same image d1(x) in Γ . ∗ So, there exists some gx ∈ G such that

d1 j(x) = gxj d1(x) . ∗ Set g∗ = 1. Let ψ : FC(X, ∗) → G be the continuous homomorphism determined by ψ(x)= gx (x ∈ X). Deﬁne a map ∗ θ : ΥC(Γ, T ) = FC(X, ∗) × Γ −→ Γ by

θ r ,m = ψ r j(m) r ∈ FC(X, ∗), m ∈ Γ .

Then θ is a graph morphism and θ(u0) = (z, s0) (the continuity of θ follows from the continuity of ψ and the continuity of the action of G∗ on Γ ∗). Put α = α θ; then α is a lifting morphism of β with α(u0) = s0.

Corollary 3.5.4 Let ζ : Γ˜ → Γ be a universal C-covering of a ﬁnite connected graph Γ , and let T be a maximal subtree of Γ . Choose a fundamental 0-section 3.5 Existence of Universal Coverings 87

: → ˜ : → C j Γ Γ of ζ lifting T , and let χj Γ π1 (Γ ) be the corresponding map (see Eq.(3.1)). Then the pointed space (χj (Γ ), 1) with distinguished point 1 is a basis C C for the free pro- group π1 (Γ ).

Proof Since the universal covering graph Γ˜ of Γ is unique up to isomorphism, we may replace ζ : Γ˜ → Γ with the explicit construction 3.5.1

υ : ΥC(Γ, T ) → Γ. ¯ ¯ First observe that the function j : Γ → ΥC(Γ, T ) given by j(m)= (1,m) (m ∈ Γ ) : → C is a fundamental 0-section, and its corresponding map χj¯ Γ π1 (Γ ) coincides with the map : → ∗ = ∗ ⊆ ∗ = C ω Γ (Γ /T , ) (X, ) FC(X, ) π1 (Γ ), so that for this choice of fundamental 0-section the result holds. Now, let e be = ∈ ∗ = C = an edge of T . Say j(e) (α, e), where α FC(X, ) π1 (Γ ). Then d0j(e) (α, d0(e)) and d1j(e) = (α, d1(e)). Since the tree T is ﬁnite, one easily deduces = ¯ = −1 = −1 ∈ that j αj. Hence χj (m) αχj¯(m)α αω(m)α , for all m Γ . Therefore = ∗ −1 ∗ = C (χj (Γ ), 1) α(X, )α , which is a basis for FC(X, ) π1 (Γ ), as asserted.

Corollary 3.5.5 Let Γ be a finite graph and let C be a pseudovariety of finite groups. Then Γ is a tree if and only if it is C-simply connected as a profinite graph.

Proof In one direction this was proved in Proposition 3.3.3(a). Suppose now that C = Γ is simply connected and let T be a maximal subtree of Γ . Since π1 (Γ ) 1, it follows from Proposition 3.5.3(b) that Γ = T .

Recall that a pro-C group G is C-projective if it is a projective object in the category of pro-C groups, i.e., if for every pair of epimorphisms of pro-C groups α : A → B and ϕ : G → B, there exists a continuous homomorphism ϕ¯ : G → A such that αϕ¯ = ϕ:

G ϕ¯ ϕ

A B α

A proﬁnite group is called projective if it is C-projective for the pseudovariety of all ﬁnite groups C.

Proposition 3.5.6 (See also Theorem 3.7.4) Let Γ be a connected profinite graph. Then (a) there exists a universal Galois C-covering ζ : Γ˜ → Γ of Γ ; C C C (b) the fundamental pro- group π1 (Γ ) of Γ is a -projective profinite group. 88 3 The Fundamental Group of a Profinite Graph

Proof Express Γ as an inverse limit

lim←− Γi i∈I of ﬁnite connected graphs Γi . By Proposition 3.5.3, for each i ∈ I there exists a ˜ universal Galois C-covering ζi : Γi → Γi of Γi . Then the results follow from Propo- sition 3.3.2, since the inverse limit of free pro-C groups is C-projective.

Proposition 3.5.7 Let Γ be a connected ﬁnite graph and let be a connected C C subgraph of Γ . Then π1 () is a free factor of π1 (Γ ).

Proof Choose a maximal subtree D of and extend it to a maximal subtree T C = − C = − of Γ . Then by Proposition 3.5.3 π1 () F( D) and π1 (Γ ) F(Γ T),so the result follows.

The construction described above of the graph ΥC(Γ, T ) parallels the construction of the abstract standard tree Υ abs(Γ, T ) of an abstract connected graph Γ (see Appendix A.2 or Theorem I.12 in Serre 1980). Let Γ be a ﬁnite connected abstract graph and let T be a maximal subtree of Γ ; consider the abstract free group Φ = Φ(Γ − T)on the set Γ − T , and construct an abstract graph

Υ abs = Υ abs(Γ, T ) = Φ(Γ − T)× Γ with set of vertices V(Υabs) = Φ(Γ − T)× V(Γ) and set of edges E(Γ ) = Φ(Γ − T)× E(Γ ); deﬁne the incidence maps d0 and d1 by

d0(f, e) = f,d0(e) and d1(f, e) = fρ(e), d1(e) ,(e∈ Γ,f ∈ Φ) where the map ρ : Γ → Φ is given by 1, if e ∈ T ; ρ(e)= e, if e ∈ E(Γ − T).

abs abs = − Then the abstract fundamental group π1 (Γ ) of Γ is π1 (Γ ) Φ(Γ T). It is useful to keep in mind the relationship between the abstract and pro-C constructions of fundamental groups and universal coverings. We make this explicit in the following proposition.

Proposition 3.5.8 Let Γ be a ﬁnite connected graph and let T be a maximal subtree of Γ . Then C C C (a) the fundamental pro- group π1 (Γ ) of Γ is the pro- completion

C = abs = − = − π1 (Γ ) π1 (Γ ) Cˆ Φ(Γ T) Cˆ FC(Γ T) abs − of the abstract fundamental group π1 (Γ ) of Γ [here FC(Γ T)denotes the free pro-C group with basis Γ − T ]; 3.6 Subgroups of Fundamental Groups of Graphs 89

(b)

= \ = ∩ abs \ abs ; ΥC(Γ, T ) lim←− N ΥC(Γ, T ) lim←− N π1 (Γ ) Υ (Γ, T ) C C N oπ1 (Γ ) N oπ1 (Γ )

(c) Υ abs(Γ, T ) is canonically embedded as a dense subgraph of ΥC(Γ, T ); abs C abs (d) π1 (Γ ) consists of the elements of π1 (Γ ) that leave Υ (Γ, T ) invariant; (e) Υ abs(Γ, T ) is an abstract connected component of ΥC(Γ, T ), where ΥC(Γ, T ) is viewed as an abstract graph; (f) there exists a fundamental 0-transversal J of ν : ΥC(Γ, T ) → Γ such that J ⊆ Υ abs(Γ, T ), and the corresponding function χ (see Eq.(3.1)) has its values in abs ∈ abs ∈ π1 (Γ ), i.e., χ(m) π1 (Γ ) for all m Γ .

Proof Parts (a)Ð(d) are clear. To prove part (e), suppose on the contrary that e is an edge of ΥC(Γ, T ) − Υ abs(Γ, T ) with one of its vertices v ∈ Υ abs(Γ, T ),say = = abs v d0(e ) (if v d1(e ), the argument is similar). Let J be a π1 (Γ )-transversal in abs = C ∈ C = Υ (Γ, T ). Since ΥC(Γ, T ) π1 (Γ )J , there exists a g π1 (Γ ) such that ge e , where e is an edge of J .Letw = d0(e); then gw = v. Since v and w have the same ∈ abs = −1 image in Γ , there exists a g π1 (Γ ) with g w v. Hence g g stabilizes w and −1 = C = therefore g g 1, because π1 (Γ ) acts freely on ΥC(Γ, T ). Hence g g , and so e ∈ Υ abs(Γ, T ), a contradiction. Finally, part (f) follows from Lemma 3.4.3 (using the notation of that lemma, just let Λ consist of a vertex of Υ abs(Γ, T ) that lifts a vertex of T ), and the deﬁnition of χ.

3.6 Subgroups of Fundamental Groups of Graphs

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

Proposition 3.6.1 Let ζ : Γ˜ → Γ be a universal Galois C-covering of a connected C proﬁnite graph Γ and let H be a closed subgroup of π1 (Γ ). (a) The canonical epimorphism

ξ : Γ˜ −→ H\Γ˜

is a universal Galois C-covering. C \ ˜ = (b) π1 (H Γ) H .

Proof Part (b) follows from (a). We prove (a) in three steps.

C Step 1. Assume that Γ is ﬁnite and H is an open subgroup of π1 (Γ ). 90 3 The Fundamental Group of a Proﬁnite Graph

Put = H \Γ˜ and let

= C : r π1 (Γ ) H C ˜ (the index of H in π1 (Γ )). It follows from the construction of Γ that is a ﬁnite graph and ||=r|Γ | and V() = rV(Γ). Let T and D be maximal subtrees of Γ and , respectively. Since the number of vertices in a ﬁnite tree equals the number of its edges plus one, we have ||−|D|=r|Γ |−2rV(Γ) + 1 = r |Γ |−2V(Γ) + 1

= r |Γ |− |T |+1 + 1 = r |Γ |−|T |−1 + 1.

Hence, by Proposition 3.5.3,

C =| |−| |= C − + rank π1 () D r rank π1 (Γ ) 1 1. = C − + On the other hand, rank(H ) r(rank(π1 (Γ )) 1) 1 (cf. RZ, Theorem 3.6.2(b)). Therefore,

= C rank(H ) rank π1 () . Let ˜ → be a universal Galois C-covering of . Since the natural morphism of ˜ ˜ graphs Γ → H \Γ = is a Galois C-covering, the identity morphism id : → lifts to a qmorphism α : ˜ → Γ˜ . By Proposition 3.1.4, there exists a continuous : C → homomorphism f π1 () H compatible with α. By Proposition 3.1.5, f is C an epimorphism. Since π1 () and H are free proﬁnite groups with the same ﬁnite rank, we deduce that f is an isomorphism (see Sect. 1.3). From Proposition 3.1.5(b) it follows that α is an isomorphism of graphs.

C Step 2. Assume that Γ is ﬁnite and H is a closed subgroup of π1 (Γ ). U C Let be the collection of all open subgroups of π1 (Γ ) containing H .Bythe ˜ ˜ case above, the canonical epimorphism ξU : Γ → U\Γ is a universal Galois C- covering (U ∈ U). Since = ξ lim←− ξU , U∈U it follows from Proposition 3.3.2 that ξ is a universal Galois C-covering.

Step 3. General case. Express Γ as an inverse limit

= Γ lim←− Γi i∈I 3.7 Universal Coverings and Simple Connectivity 91 of ﬁnite graphs Γi (see Proposition 2.1.4). By Proposition 3.3.2, the universal Galois C-coverings ˜ {ζi : Γi −→ Γi | i ∈ I} form an inverse system such that = ζ lim←− ζi i∈I and C = C π1 (Γ ) lim←− π1 (Γi). i∈I C C → C Denote by Hi the image of H on π1 (Γi) under the projection π1 (Γ ) π1 (Γi). ˜ ˜ Clearly the canonical epimorphisms ξi : Γi → Hi\Γi form an inverse system and = ξ lim←− ξi. i∈I

By Step 2, ξi is a universal Galois C-covering. Thus, by Proposition 3.3.2, ξ is a universal Galois C-covering.

3.7 Universal Coverings and Simple Connectivity

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

Theorem 3.7.1 (a) Let ζ : Γ → be a Galois C-covering. Then ζ is universal if and only if Γ is C-simply connected. (b) Let Γ be a proﬁnite graph. Then the identity morphism id : Γ → Γ is a universal Galois C-covering if and only if Γ is C-simply connected. (c) Let Γ be a C-simply connected proﬁnite graph. Then any Galois C-covering ζ : Σ → Γ is trivial, i.e., G(Σ|Γ)= 1.

: → = C N Proof (a) Assume ﬁrst that ζ Γ is universal and let G π1 ().Let be the collection of all closed normal subgroups N of G. According to Proposi- : → \ C \ = tion 3.6.1, the natural qmorphism ζN Γ N Γ is universal with π1 (N Γ) N. Since = ζ lim←− ζN , N∈N we have (see Proposition 3.3.2(b)) C = C \ = = π1 (Γ ) lim←− π1 (N Γ) N 1. N∈N N∈N 92 3 The Fundamental Group of a Proﬁnite Graph

C C = : Conversely, suppose that Γ is -simply connected, i.e., that π1 (Γ ) 1. Let ξ ˜ → be a universal Galois C-covering. By the universal property of ξ, there exists a qmorphism of proﬁnite graphs α : ˜ → Γ such that ζα= ξ. By Proposition 3.1.5, α is a Galois covering and G(˜ |Γ) is a closed subgroup of G(˜ |). It follows from Proposition 3.6.1 that α is a universal Galois C-covering. Since Γ is C-simply ˜ | = C = ˜ = connected, G( Γ) π1 (Γ ) 1; therefore, Γ , as required. (b) Follows from (a). (c) By part (b) the identity morphism id : Γ → Γ is a universal Galois C- covering. Let α : Γ → Σ be such that ζα= id. Then by Proposition 3.1.5, G(Σ|Γ) is a quotient of G(Γ |Γ)= 1, and so G(Σ|Γ)= 1.

Corollary 3.7.2 Let ζ : → Γ be a connected Galois C-covering and let T be a C-simply connected proﬁnite subgraph of Γ . (a) Let m ∈ such that ζ(m ) = m ∈ T . Then there exists a unique lifting T of T in such that m ∈ T . (b) If T and T are liftings of T in , there exists a unique g ∈ G(ζ ) such that T = gT . (c) Assume that T is in fact a spanning C-simply connected proﬁnite subgraph of Γ . Then there exists a fundamental 0-transversal J in lifting T . If J is another such fundamental 0-transversal, then there exists a unique g ∈ G(ζ ) such J = gJ .

Proof (a) By Theorem 3.7.1(b), id : T → T is universal. Therefore there exists a morphism τ : T → such that ζτ = idT and τ(ζ(m))= m. Then T = τ(T) is the desired lifting. Note that τ is unique by Lemma 3.1.7. (b) Let m ∈ T and m ∈ T with ζ(m ) = ζ(m ) = m. Then there exists a unique g ∈ G(ζ ) such that m = gm . Since m ∈ gT and gT is also a lifting of T , it follows from (a) that T = gT . (c) This follows from parts (a) and (b) and Lemma 3.4.3.

Proposition 3.7.3 Let be a connected profinite subgraph of a connected profinite graph Γ and assume that ζ : Γ˜ → Γ and ξ : ˜ → are universal Galois C-coverings. Then we have the following properties. (a) There exists an injective lifting morphism ψ˜ : ˜ → Γ˜ of the natural embedding ψ : → Γ . C C (b) π1 () is isomorphic to a closed subgroup of π1 (Γ ). ˜ −1 C (c) can be identified with a connected component of ζ (), and π1 () with C the maximal subgroup of π1 (Γ ) that stabilizes that component. (d) A connected profinite subgraph of a C-simply connected profinite graph is C- simply connected.

Proof Part (d) is an immediate consequence of (b). To prove the other parts, assume first that Γ is finite. Choose maximal trees D and T of and Γ , respectively, so that T ∩ = D. By Propositions 3.5.3 and 3.3.1(b) we may identify ˜ and Γ˜ 3.7 Universal Coverings and Simple Connectivity 93 with (the canonically constructed graphs) ΥC(, D) and ΥC(Γ, T ), respectively, and we may assume that ξ and ζ are the natural projections. By Proposition 3.5.3, C = − C = − π1 () FC( D) is a subgroup of π1 (Γ ) FC(Γ T). From the constructions of ΥC(Γ, T ) and ΥC(, D) (see 3.5.1) it follows that ΥC(, D) is a profinite subgraph of ΥC(Γ, T ). − ∗ We claim that ΥC(, D) is a connected component of ζ 1(). To see this let G C be the maximal subgroup of π1 (Γ ) which leaves invariant the connected component − ∗ − (ζ 1()) of ζ 1() containing ΥC(, D). The restriction

: −1 ∗ −→ ζ|(ζ −1())∗ ζ () is a Galois C-covering with associated group G∗: indeed, since (ζ −1())∗ is a −1 maximal connected proﬁnite subgraph of ζ (),wehavethatgm1 = m2 implies ∈ ∗ ∈ −1 ∗ ∈ C g G , whenever m1,m2 (ζ ()) , g π1 (Γ ) (see Lemma 2.1.7). Then, by Proposition 3.1.5, the natural embedding

−1 ∗ ΥC(, D) −→ ζ () is surjective and so it is an isomorphism. This proves the claim, and also parts (a), (b) and (c) when Γ is finite. Now suppose Γ is infinite. Write Γ as an inverse limit = Γ lim←− Γi i∈I of an inverse system {Γi,ϕij ,I} of finite graphs. Then = lim←− i, i∈I where i is the image of in Γi under the projection ϕi : Γ → Γi . Choose a point m ∈ and denote by mi its image in i (i ∈ I ). Denote by ˜ ˜ ζi : Γi −→ Γi and ξi : i −→ i the universal Galois C-coverings of Γi and i , respectively. By the finite case con- ˜ sidered above, i can be identified canonically with a connected component of −1 ∈ ˜ ∈ −1 ∈ ζi (i) (i I ). Choose mi ξi (mi) for every i I . By Proposition 3.3.2, there ˜ ˜ ˜ exists an inverse system {Γi, ϕ˜ij ,I}, where ϕ˜ij : Γi → Γj is the (unique) lifting of ϕij : Γi → Γj such that ϕ˜ij (m˜ i) =˜mj (i, j ∈ I,i j); moreover,

˜ = ˜ Γ lim←− Γi. i∈I ˜ ˜ Then clearly ϕij (i) ⊆ j ; furthermore ϕ˜ij (i) ⊆ j , because ϕ˜ij maps the ˜ −1 ˜ connected component of mi in ζi (i) into the connected component of mj in −1 : → ˜ : ˜ → ˜ ζi (j ). Now, denote by ρij i j and ρij i j the restrictions of ϕij 94 3 The Fundamental Group of a Proﬁnite Graph

˜ ˜ and ϕ˜ij to i and i , respectively (i, j ∈ I,i j). Therefore, {i, ρ˜ij ,I} is an inverse system and, by Proposition 3.3.2, ˜ = ˜ lim←− i. i∈I ˜ −1 ˜ ˜ Since i is the connected component of ζi (i) containing mi , we have that is the connected component of ζ −1(Γ ) containing m˜ . This proves parts (a) and (c). Part (b) follows from the ﬁnite case above and Proposition 3.3.2.

In contrast with the situation for abstract graphs, the fundamental pro-C group C C π1 (Γ ) of a connected profinite graph Γ is not always free pro- . In fact, we show below (Theorem 3.8.3) that every projective pro-C group can be realized as the fun- C C damental pro- group π1 (Γ ) of some connected profinite graph Γ . In the next theorem we give a sufficient condition for a connected graph Γ to have a free fundamental pro-C group.

Theorem 3.7.4 Let Γ be a connected proﬁnite graph having a spanning C-simply connected proﬁnite subgraph T . Then the following properties hold.

(a) The Galois C-covering υ : ΥC(Γ, T ) → Γ constructed in 3.5.1 is universal. C C ∗ = ∗ (b) π1 (Γ ) is a free pro- group on the pointed proﬁnite space (X, ) (Γ /T , ).

Proof Part (b) follows from (a) and the construction 3.5.1. To prove (a) let ζ : Γ˜ → Γ be a universal C-covering of Γ . Choose a fundamental 0-transversal J in Γ˜ lifting T (see Corollary 3.7.2). Denote by j : Γ → J the corresponding fundamental 0-section. Recall that j|T : T → j(T)= T is an isomorphism of proﬁnite graphs. We shall construct an isomorphism α : ΥC(Γ, T ) → Γ˜ such that ζα= υ.

α ΥC(Γ, T ) Γ˜

υ ζ Γ

: → C Consider the continuous function χ Γ π1 (Γ ) (see Eq. (3.1) in Sect. 3.4)given by the formula

d1 j(m) = χ(m)j d1(m) (m ∈ Γ). (3.2)

We use the notation of 3.5.1, in particular recall that ΥC(Γ, T ) = FC(X, ∗) × Γ , where the pointed space X is (X, ∗) = (Γ /T , ∗); and

ω : Γ → (X, ∗) = (Γ /T , ∗) is the canonical projection. Since ω sends every t ∈ T to ∗ and χ sends t ∈ T to 1, χ induces a continuous map of pointed spaces (that we still denote by χ) 3.8 Fundamental Groups and Projective Groups 95

: ∗ → C χ (X, ) π1 (Γ ).Let : ∗ −→ C θ FC(X, ) π1 (Γ ) be the continuous homomorphism extending χ. Deﬁne a map

α : ΥC(Γ, T ) = FC(X, ∗) × Γ −→ Γ,˜ by

α(f,m) = θ(f)j(m) f ∈ FC(X, ∗), m ∈ Γ . Using equality (3.2) one easily checks that α is a qmorphism of proﬁnite graphs. Clearly ζα = υ. By Proposition 3.1.5, α is surjective. It then follows from the C- simple connectivity of Γ˜ that α and θ are isomorphisms (see Theorem 3.7.1 and Proposition 3.1.5).

3.8 Fundamental Groups and Projective Groups

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

Proposition 3.8.1 The Cayley graph Γ(FC(X, ∗), X) of a free pro-C group on a pointed proﬁnite space (X, ∗) with respect to X is C-simply connected. In fact, Γ(FC(X, ∗), X) is the universal Galois C-covering space of the bouquet of loops = ∗ C = ∗ B B(X, ) and π1 (B) FC(X, ).

Proof Consider the bouquet of loops B = B(X,∗) associated with (X, ∗) (see Ex- ample 3.1.1). Plainly the subgraph T ={∗}is a spanning C-simply connected proﬁ- nite subgraph of B and ΥC(B, T ) coincides with the Cayley graph Γ(FC(X, ∗), X). By Theorem 3.7.4, Γ(FC(X, ∗), X) is the universal Galois C-covering graph of B, C = ∗ ∗ C with π1 (B) FC(X, ). So, by Theorem 3.7.1, Γ(FC(X, ), X) is -simply connected.

Corollary 3.8.2 Let C ⊆ C be extension-closed pseudovarieties of ﬁnite groups. Let (X, ∗) be a pointed topological space and consider the natural morphism of graphs

γ : Γ FC(X, ∗), X −→ Γ FC (X, ∗), X , where F = FC(X, ∗) (respectively, F = FC (X, ∗)) is the free pro-C (respectively, pro-C ) group on the pointed space (X, ∗). Then γ is a universal Galois C-covering map with fundamental group

C ∗ = ∗ −→ ∗ π1 Γ FC (X, ), X Ker FC(X, ) FC (X, ) . 96 3 The Fundamental Group of a Proﬁnite Graph

Proof Let B = B(X,∗) be the bouquet of loops associated with (X, ∗) (see Exam- ple 3.1.1). We have a commutative diagram

γ Γ(FC(X, ∗), X) Γ(FC (X, ∗), X)

ζ ζ B where ζ and ζ are quotient maps modulo F and F , respectively. As pointed out in Proposition 3.8.1, ζ and ζ are the universal Galois C-covering and C -covering of B, respectively. By Theorem 3.7.1(a), Γ(FC(X, ∗), X) is C-simply connected. By Proposition 3.1.5, γ is a Galois covering whose associated group is the kernel of the epimorphism

FC(X, ∗) −→ FC (X, ∗). The result then follows from Theorem 3.7.1(a) again.

Recall that (under our standing assumption that C is an extension-closed pseudovariety of finite groups) a pro-C group G is C-projective if and only if it is a closed subgroup of a free pro-C group, or equivalently, if and only if it is projective, i.e., if and only if it is a closed subgroup of a free profinite group (see Sect. 1.5). The next theorem gives a profinite graph-theoretic characterization of projective pro-C groups.

Theorem 3.8.3 A profinite group G is projective if and only if there exists a con- C =∼ nected profinite graph Γ such that π1 (Γ ) G, for some extension-closed pseudovariety of finite groups C.

C =∼ Proof If π1 (Γ ) G, then G is projective by Proposition 3.5.6 and the above comment. Conversely, assume that G is projective. Then G is a closed subgroup of a free profinite group F(X,∗) on some pointed profinite space (X, ∗). Then G is the fundamental profinite group of the quotient graph G\Γ(F(X,∗), X) of the Cayley graph Γ(F(X,∗), X) (see Propositions 3.6.1 and 3.8.1).

3.9 Fundamental Groups of Quotient Graphs

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

In the next two propositions we show that certain types of epimorphisms of connected proﬁnite graphs preserve C-simple connectivity. And, more generally, in some cases they preserve fundamental groups. 3.9 Fundamental Groups of Quotient Graphs 97

Proposition 3.9.1 Let be a profinite subgraph of a connected profinite graph Γ such that every connected component of is C-simply connected. Let Γ be the profinite quotient graph of Γ obtained by collapsing each connected component of to a point (see Exercise 2.1.11). Then C = C π1 (Γ ) π1 (Γ).

In particular, if Γ is C-simply connected, so is Γ.

Proof Assume first that Γ is C-simply connected. In light of Theorem 3.7.1,we need to show that : −→ idΓ Γ Γ is a universal Galois C-covering. To see this it is necessary to show that for an arbitrary qmorphism of connected profinite graphs β : Γ → Ω, an arbitrary connected Galois C-covering ξ : Σ → Ω and any m ∈ Γ,s ∈ Σ with β(m) = ξ(s), there exists a qmorphism of profinite graphs α : Γ → Σ such that β = ξα and α(m) = s.

ω Γ Σ α τ ξ

Γ Ω β

−1 Let τ : Γ → Γ be the natural epimorphism and let m0 ∈ τ (m). Since Γ is C-simply connected, there exists a qmorphism of proﬁnite graphs ω : Γ → Σ such ∗ that ξω = βτ and ω(m0) = s. We claim that the image ω( ) of any connected component ∗ of consists of a single point, i.e., one vertex. Since ω(∗) is connected, this is equivalent to showing that E(ω(∗)) =∅. Suppose not, and let e ∈ E(∗) be such that ω(e) ∈ E(ω(∗)). Then

ξω(e)= βτ(e) = β(v) ∈ V(Ω),

∗ where v is the vertex of Γ obtained from collapsing . However, since ξ is a Ga- lois C-covering, ξ(E(Σ)) ⊆ E(Ω). This is a contradiction, and therefore the claim is established. Thus the map

α : Γ −→ Σ −1 given by α(m) = ω(τ (m)) (m ∈ Γ) is a well-defined qmorphism of profinite graphs. Obviously α satisfies the desired properties. Now let Γ be an arbitrary connected profinite graph and let ζ : Γ˜ → Γ be a universal Galois C-covering. Let Θ denote the graph obtained by collapsing the connected components of ζ −1(). By the case above, the graph Θ is C-simply con- C ˜ C nected. The free action of π1 (Γ ) on Γ induces an action of π1 (Γ ) on Θ (since 98 3 The Fundamental Group of a Profinite Graph

−1 C C ζ () is π1 (Γ )-invariant and the action of an element of π1 (Γ ) sends a connected component to a connected component). The stabilizers of the connected components of ζ −1() are isomorphic to the fundamental groups of the connected components of (see Proposition 3.7.3). Therefore these stabilizers are trivial since, by assumption, the connected components of are C-simply connected. Thus the C C π1 (Γ )-stabilizers of all vertices of Θ are trivial, i.e., π1 (Γ ) acts freely on Θ. Since Θ is C-simply connected, −→ C \ Θ π1 (Γ ) Θ is a Galois C-covering which is universal by Theorem 3.7.1. Clearly Γ = C \ π1 (Γ ) Θ. So, C = C π1 (Γ ) π1 (Γ), as desired.

Proposition 3.9.2 Let G be a pro-C group acting on a C-simply connected proﬁnite graph Γ . Suppose that G is generated by the stabilizers of the elements of Γ :

G = Gm | m ∈ Γ .

Then the quotient graph G\Γ is C-simply connected.

Proof By Theorem 3.7.1, it is sufﬁcient to check that

idG\Γ : G\Γ −→ G\Γ is a universal Galois C-covering. Let β : G\Γ → be a qmorphism of proﬁnite graphs and let ξ : Σ → be a Galois C-covering. Choose x ∈ G\Γ,s ∈ Σ with β(x) = ξ(s). To check the universal property of idG\Γ we need to construct a morphism α : G\Γ −→ Σ such that ξα = β and α(x) = s.Letτ : Γ → G\Γ be the natural epimorphism −1 and let x0 ∈ τ (x). Since Γ is C-simply connected, we have that idΓ is universal; hence, there exists a morphism ω : Γ → Σ such that ξω= βτ and ω(x0) = s.

ω Γ Σ α τ ξ

G\Γ β

We claim that the image ω(Gm0) of the G-orbit of an element m0 ∈ Γ consists of exactly one element, namely ω(m0). By continuity it sufﬁces to show that if m ∈ Γ 3.9 Fundamental Groups of Quotient Graphs 99 and gm ∈ Gm , then

ω(gm m0) = ω(m0). Note that

ξ ω(gm m0) = βτ(gm m0) = βτ(m0) = ξ ω(m0) .

Therefore, ω(gm m0) = gω(m0),forsomeg ∈ G(Σ|). We shall show that g = 1. Denote by δ and γ the automorphisms of Γ and Σ deﬁned by multiplication by gm and g, respectively.

ω Γ Σ

δ γ ω Γ Σ α τ ξ

G\Γ β

Consider the morphisms ωδ,γ ω : Γ → Σ. One easily checks that ξωδ = ξγω; moreover, by the deﬁnition of g, ωδ(m0) = γω(m0). Hence, by Lemma 3.1.7, ωδ = γω.

It follows that

ω m = ω gm m = ωδ m = γω m = gω m . By the freeness of the action of G(Σ|) on Σ,wehaveg = 1, as desired. This proves the claim. Thus, one can deﬁne α : G\Γ −→ Σ by α(m) = ω(τ−1(m)). Clearly α satisﬁes the desired properties.

Corollary 3.9.3 Let G be a pro-C group acting on a C-simply connected proﬁnite graph Γ . Let

N = Gm | m ∈ Γ be the closed subgroup of G generated by the stabilizers Gm of the elements m ∈ Γ . Then N is normal in G and C \ =∼ π1 (G Γ) G/N. In particular, the group G/N is projective. Consequently, if G acts freely on Γ , then G is projective. 100 3 The Fundamental Group of a Proﬁnite Graph

−1 Proof The normality of N follows from the equality gGmg = Ggm. Plainly G/N acts freely on N\Γ ; so the natural epimorphism ζ : N\Γ −→ G\Γ = (G/N)\(N\Γ) is a Galois C-covering with associated group G((N\Γ)| (G\Γ))= G/N.By Proposition 3.9.2, N\Γ is C-simply connected. Therefore, by Theorem 3.7.1, ζ is C \ = universal. This means that π1 (G Γ) G/N. Finally, G/N is projective by Theo- rem 3.8.3.

3.10 π-Trees and Simple Connectivity

In this section we study the relationship between the notions of π-tree and simple connectivity. For the concept of complete tensor product of proﬁnite modules that we use in the next theorem, see Sect. 1.8.

Theorem 3.10.1 Let C be a pseudovariety of ﬁnite groups which is closed under extensions with abelian kernel. Assume that Γ is a connected proﬁnite graph and let ζ : Γ˜ → Γ be its universal Galois C-covering graph. Then Γ˜ is a C-tree.

Proof Case 1. Γ is a ﬁnite graph. Let T be a maximal subtree of Γ . Put X = Γ − T , the set of edges of Γ not in T , and let F = FC(X) be the free pro-C group on the set X. In this case Γ˜ (see 3.5.1 and Proposition 3.5.3) can be described as having vertices V(Γ)˜ = F × V(Γ)= F × V(T), edges

E(Γ)˜ = F × E(Γ ) = (F × X)∪. F × E(T ) ˜ ˜ and incidence maps d0,d1 : E(Γ) → V(Γ) deﬁned as follows: d0(f, x) = (f, d0(x)), d1(f, x) = (f x, d1(x)), for all f ∈ F , x ∈ X, and d0(f, e) = (f, d0(e)), d1(f, e) = (f, d1(e)), for all f ∈ F , e ∈ E(T ). One easily sees that, in this case, to show that Γ˜ is a C-tree we have to prove that the sequence

˜ d ˜ ε 0 −→ ZCÊ(Γ) −→ ZCˆV(Γ) −→ ZCˆ −→ 0 (3.3) is exact, where d is the continuous homomorphism of free profinite ZCˆ-modules ˜ ˜ that restricts to d1 − d0 on E(Γ), and ε is the map that sends each v ∈ V(Γ) to 1 ∈ ZCˆ. Since T is a finite tree we know that

−→ d ε 0 ZCˆE(T ) −→ ZCˆV(T) −→ ZCˆ −→ 0 (3.4) is exact. 3.10 π-Trees and Simple Connectivity 101

For a proﬁnite space Y , let us denote the kernel of the augmentation homomorphism ε :[[ZCˆY ]] → ZCˆ by (( I Y )) . Clearly

[[ ZCˆY ]] = (( I Y )) ⊕ ZCˆ.

One check that if Y1 and Y2 are proﬁnite spaces, then

ZCˆ(Y1 × Y2) =[[ZCˆY1]] ⊗[[ZCˆY2]] . It follows that

˜ ZCˆE(Γ) = ZCˆ(F × X) ⊕ ZCˆ F × E(T ) and

˜ ZCˆV(Γ) =[[ZCˆF ]] ⊗ ZCˆV(T) = (( I F )) ⊕ ZCˆ ⊗ IV(T) ⊕ ZCˆ

= (( I F )) ⊗ ZCˆ ⊕ (ZCˆ ⊗ ZCˆ) ⊕ [[ ZCˆF ]] ⊗ IV(T)

= (( I F )) ⊕ ZCˆ ⊕ [[ ZCˆF ]] ⊗ IV(T) . (3.5)

Next observe that the restriction of d to [[ ZCˆ(F × X)]] is the map

ZCˆ(F × X) → (( I F )) that sends (f, x) to fx− f , which is an isomorphism as one easily deduces from Lemma 2.5.2. On the other hand, the restriction of d to [[ ZCˆ(F × E(T ))]] may be identiﬁed with the homomorphism

id⊗ˆ d [[ ZCˆF ]] ⊗ ZCÊ(T ) −→ [[ ZCˆF ]] ⊗ ZCˆV(T) , ˆ where d(e) = d1(e) − d0(e),fore ∈ E(T ). Now note that id⊗d is injective because it is obtained by tensoring (3.4) with [[ ZCˆF ]] , and this is a free ZCˆ-module (see ˜ Sect. 1.8). This proves that the sequence (3.3) is exact at [[ ZCÊ(Γ)]] . ˜ To prove exactness at [[ ZCˆV(Γ)]] first note that from the above considerations and the exactness of (3.4)wehave

˜ d ZCˆE(Γ) = (( I F )) ⊕ [[ ZCˆF ]] ⊗ IV(T) ; ˜ and, from the description of [[ ZCˆV(Γ)]] in (3.5), one has that this coincides with Ker(ε), as needed.

Case 2. General connected proﬁnite graph Γ .

Write Γ as an inverse limit of its ﬁnite quotient graphs Γi (see Proposition 2.1.4). ˜ ˜ Then Γ is the inverse limit of the universal C-covering graphs Γi (see Proposi- ˜ ˜ tion 3.3.2). By Case 1, each Γi is a C-tree, and therefore so is Γ by Proposi- tion 2.4.3(d). 102 3 The Fundamental Group of a Proﬁnite Graph

Corollary 3.10.2 Let C be an extension-closed pseudovariety of ﬁnite groups. Then every C-simply connected proﬁnite graph Γ is a C-tree.

Proof By Theorem 3.7.1(b), Γ is its own universal Galois C-covering graph. Hence the result is a consequence of Theorem 3.10.1.

In contrast to the corollary above, see Example 3.10.6. Let R be a profinite ring and let G be a profinite group. Recall that a profinite G-space X is a profinite space on which G acts continuously. For a G-space X,the free R-module [[ RX]] with basis X naturally becomes a profinite [[ RG]] -module, where [[ RG]] is the complete group algebra. Similarly, if (X, ∗) is a pointed profinite G-space, then the free profinite R-module [[ R(X,∗)]] is naturally a profinite [[ RG]] - module.

Lemma 3.10.3 Let G be a profinite group and let R be a commutative profinite ring. Assume that B is an R-module, which we view as an [[ RG]] -module with trivial G action. (a) If X is a profinite G-space, there exists an isomorphism of complete tensor products

∼ B ⊗[[ RG]] [[ RX]] = B ⊗R R(G\X) . (b) If (X, ∗) is a pointed proﬁnite G-space, there exists an isomorphism of complete tensor products

∼ B ⊗[[ RG]] R(X,∗) = B ⊗R R(G\X, ∗) .

Proof The proofs of parts (a) and (b) are similar; we only prove (a). Write X as an inverse limit X = lim X ←− i of ﬁnite G-spaces X (see Sect. 1.3). Then G\X = lim(G\X ). Correspondingly i ←− i one has decompositions ∼ B ⊗[[ ]] [[ RX]] = lim B ⊗[[ ]] [[ RX ]] RG ←− RG i and

B ⊗ R(G\X) =∼ lim B ⊗ R(G\X ) . R ←− R i Hence it sufﬁces to prove the lemma when X is ﬁnite. In that case the complete tensor product coincides with the usual tensor product (cf. RZ, Proposition 5.5.3(d)) and the result becomes

∼ B ⊗[[ RG]] [RX] = B ⊗R R(G\X) . 3.10 π-Trees and Simple Connectivity 103

To verify this, consider the map ϕ : B × X → B ⊗R [R(G\X)] given by ϕ(b,x) = b ⊗ Gx (b ∈ B,x ∈ X). This map extends naturally to a middle G-linear map ϕ : B ×[RX]→B ⊗R [R(G\X)], and therefore it induces a homomorphism

ϕ : B ⊗[[ RG]] [RX]−→B ⊗R R(G\X) .

This homomorphism is an isomorphism because, as one easily checks, it has an inverse induced by the map

B × G\X −→ B ⊗[[ RG]] [RX] given by (b, Gx) → b ⊗ x(b∈ B,x ∈ X).

Proposition 3.10.4 Let Γ be a proﬁnite connected graph and let C be a pseudovariety of ﬁnite groups which is closed under extensions with abelian kernel. Then C (a) H1(Γ, ZCˆ) is the abelianized group of π1 (Γ ):

=∼ C =∼ C C C H1(Γ, ZCˆ) H1 π1 (Γ ), ZCˆ π1 (Γ )/ π1 (Γ ), π1 (Γ ) . (b) If C consists of solvable groups, then every C-tree is C-simply connected.

Proof Set π = π(C). Since C is closed under extensions with abelian kernel, one = = has ZCˆ Zπˆ p∈π Zp. (a) Let ζ : Γ˜ → Γ be a universal Galois C-covering. By Theorem 3.10.1, Γ˜ is a π-tree. Hence one has the following exact sequence of Zπˆ -modules

∗ ˜ d ˜ ε 0 [[ Zπˆ (E (Γ),∗)]] [[ Zπˆ (V (Γ))]] Zπˆ 0. (3.6)

= C ˜ ∗ ˜ ∗ Put G π1 (Γ ). The action of G on Γ , induces actions on the spaces (E (Γ), ) ˜ ∗ ˜ ˜ and V(Γ); this implies that [[ Zπˆ (E (Γ),∗)]] and [[ Zπˆ (V (Γ))]] are also [[ Zπˆ G]] - modules (cf. RZ, Proposition 5.3.6). We think of Zπˆ as a [[ Zπˆ G]] -module with trivial G-action. Furthermore, because of the deﬁnition of these actions and the deﬁnition of d and ε, we deduce that these maps are [[ Zπˆ G]] -linear. Therefore the sequence (3.6) is in fact a short exact sequence of [[ Zπˆ G]] -modules. Using the left derived [[ ]] { Zπˆ G − } ⊗ − functors Torn (Zπˆ , ) of the functor Zπˆ [[ Zπˆ G]] , we get a corresponding long exact sequence of Zπˆ -modules (see Sect. 1.9) [[ ]] ···→ Zπˆ G ˜ → Tor1 Zπˆ , Zπˆ V(Γ) H1(G, Zπˆ )

→ ⊗ ∗ ˜ ∗ Zπˆ [[ Zπˆ G]] Zπˆ E (Γ),

−→d ⊗ ˜ −→ε ⊗ → Zπˆ [[ Zπˆ G]] Zπˆ V(Γ) Zπˆ [[ Zπˆ G]] Zπˆ 0, ⊗ − where d and ε denote the maps obtained after applying the functor Zπˆ [[ Zπˆ G]] to the corresponding maps d and ε in the sequence (3.6). Now, since the action of G 104 3 The Fundamental Group of a Proﬁnite Graph

˜ ˜ on V(Γ)is free, [[ Zπˆ (V (Γ))]] is a free [[ Zπˆ G]] -module (cf. RZ, Proposition 5.7.1). Therefore [[ ]] Zπˆ G ˜ = Tor1 Zπˆ , Zπˆ V(Γ) 0. On the other hand, ⊗ = = Zπˆ [[ Zπˆ G]] Zπˆ H0(G, Zπˆ ) Zπˆ , since the action of G on Zπˆ is trivial (see Sect. 1.10). Thus, using Lemma 3.10.3 and the fact that G\Γ˜ = Γ , the last terms of the above exact sequence become

∗ d ε 0 −→ H1(G, Zπˆ ) −→ Zπˆ E (Γ ), ∗ −→ Zπˆ V(Γ) −→ Zπˆ −→ 0.

This implies (see the deﬁnition of H1(Γ, Zπˆ ) in Sect. 2.3) that

H1(Γ, Zπˆ ) = H1(G, Zπˆ ).

Finally, it is known (see Sect. 1.10) that

H1(G, Zπˆ ) = G/[G, G].

(b) If T is a C-tree, then H1(T , Zπˆ ) = 0. So, by part (a),

C = C C π1 (T ) π1 (T ), π1 (T ) . C C C = If consists of solvable groups, π1 (T ) is prosolvable, and thus π1 (T ) 1, i.e., T is C-simply connected.

Corollary 3.10.5 Let C be a pseudovariety of ﬁnite groups closed under extensions with abelian kernel. Let Γ be a connected proﬁnite graph. Then the following conditions are equivalent: (a) Γ is a C-tree; C C (b) the fundamental pro- group π1 (Γ ) of Γ is perfect, i.e., it coincides with its commutator subgroup.

Proof By deﬁnition, Γ is a C-tree if and only if H1(Γ, ZCˆ) is trivial. Therefore the result follows from Proposition 3.10.4(a).

The next example shows that a π-tree is not always simply connected.

Example 3.10.6 (A C-tree which is not C-simply connected) Let (X, ∗) be a pointed profinite space with |X| > 2. Let Cs be the pseudovariety of all finite solvable groups and let C be the pseudovariety of all finite groups. Then π(Cs) = π(C); put π = C = C = ∗ = ∗ π( s) π( ).LetFs FCs (X, ) and F FC(X, ) be the free prosolvable and the free profinite group on (X, ∗), respectively. Consider the Cayley graphs Γs = Γ(Fs,X)and Γ = Γ(F,X)of Fs and F with respect to X, respectively. According 3.11 Free Pro-C Groups and Cayley Graphs 105 to Theorem 2.5.3, Γs is a Cs -tree, i.e., a π-tree, or equivalently, C-tree. We claim that C C Γs is not -simply connected. Indeed, according to Corollary 3.8.2, π1 (Γs) equals the kernel of the natural continuous epimorphism = ∗ −→ = ∗ F FC(X, ) Fs FCs (X, ), which obviously is not trivial.

We conclude this section with an example that shows that the notion of a π-tree depends on the set of primes π.

Example 3.10.7 (The notion of a π-tree depends on π) Here we exhibit examples to make explicit that the notion of a π-tree depends on the choice of the set of primes π. Let p and q be different prime numbers. Let Cp be the pseudovariety of all finite p-groups, and let Cp,q be the pseudovariety of all finite groups whose order is of the form piqj , where i, j are natural numbers. Let (X, ∗) be a profinite pointed space with |X| > 1, and let

= ∗ Γ Γ FCp (X, ), X ∗ ∗ be the Cayley graph of the free pro-p group FCp (X, ) on (X, ) with respect to X. By Theorem 2.5.3, Γ is a p-tree. Cp,q By Corollary 3.8.2, π1 (Γ ) is the kernel of the natural continuous epimorphism ∗ −→ ∗ FCp,q (X, ) FCp (X, ).

C p,q = Hence π1 (Γ ) 1. On the other hand, by a well-known theorem of Burnside C C p,q (cf. Hall 1959, Theorem 9.3.2), all groups in p,q are solvable, so that π1 (Γ ) is prosolvable, and therefore nonperfect. So, according to Corollary 3.10.5, Γ is not a {p,q}-tree. Since Γ is a p-tree, it follows that it is not a q-tree (see Proposi- tion 2.4.3(e)).

3.11 Free Pro-C Groups and Cayley Graphs

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

Let X be a closed subset of a pro-C group G.NextwegiveacriterionforG to be a free pro-C group on X in terms of Cayley graphs.

Theorem 3.11.1 Let G beapro-C group and let X be a closed subset of G such that 1 ∈ X. Then G is the free pro-C group on the pointed space (X, 1) if and only if the Cayley graph Γ(G,X)of G with respect to X is C-simply connected. 106 3 The Fundamental Group of a Proﬁnite Graph

Proof Denote by μ : X → G the inclusion map. View X as a pointed space with distinguished point ∗=1. Let

f : FC(X, ∗) −→ G be the continuous homomorphism from the free pro-C group FC(X, ∗) on (X, ∗) into G induced by μ. Consider the bouquet of loops B = B(X,∗) (see Example 3.1.1). Its universal Galois C-covering is ζ : Γ(FC(X, ∗), X) → B; therefore, by Proposition 3.8.1, Γ(FC(X, ∗), X) is C-simply connected. C C = Conversely, assume that Γ(G,X)is -simply connected, i.e., π1 (Γ (G, X)) 1. Then, by deﬁnition, Γ(G,X)is a connected proﬁnite graph. [Hence G = X (see Lemma 2.2.4(c)), and so f is surjective.] Therefore ξ : Γ(G,X)→ B is a connected Galois C-covering of B with associated group G (see Example 3.1.1). Consider the commutative diagram

α Γ(FC(X, ∗), X) Γ(G,X)

ζ ξ B where α is the epimorphism of proﬁnite graphs given by α(g,x) = (f (g), μ(x)). By Proposition 3.1.5, α is a Galois C-covering whose associated group is G(α) = Ker(f ). Since Γ(FC(X, ∗), X) is C-simply connected,

α : Γ FC(X, ∗), X → Γ(G,X) is the universal Galois C-covering of Γ(G,X).So

= C = = 1 π1 Γ(G,X) G(α) Ker(f ).

Thus f is an isomorphism and G =∼ F(X,∗), as desired.

Corollary 3.11.2 Assume that all the groups in the class C are solvable. Let G be apro-C group and let X be a closed subset of G such that 1 ∈ X. Then G is a free pro-C group on the pointed space (X, 1) if and only if the Cayley graph Γ(G,X)is a C-tree.

Proof Since the groups in C are solvable, a proﬁnite graph is a C-tree if and only if it is C-simply connected (see Corollary 3.10.2 and Proposition 3.10.4(b)). Hence the result follows from Theorem 3.11.1. 3.12 Change of Pseudovariety 107

3.12 Change of Pseudovariety

In this section we study how universal Galois C-coverings, fundamental pro-C groups, etc. change when the pseudovariety C changes. Let C ⊆ C be pseudovarieties of ﬁnite groups. Let Γ be a given connected proﬁ- nite graph. Denote by

ζ : Γ˜ → Γ and ζ : Γ˜ → Γ the universal Galois C-covering and universal Galois C -covering of Γ , respectively. Then there exists an epimorphism of proﬁnite graphs γ : Γ˜ → Γ˜ making the following diagram

γ Γ˜ Γ˜

ζ ζ Γ commutative, since ζ is a Galois C -covering. In addition, there exists a continuous : C → C epimorphism f π1 (Γ ) π1 (Γ ) such that γ(gm)= f(g)γ(m), ∈ C ∈ ˜ for all g π1 (Γ ), m Γ (see Proposition 3.1.4). By Proposition 3.1.5(b) and C ˜ ˜ | ˜ = C ˜ Theorem 3.7.1(a), γ is a universal Galois -covering of Γ and G(Γ Γ) π1 (Γ) can be identified with Ker(f ). Choose a 0-transversal j of ζ and a 0-transversal jγ of γ (see Lemma 3.4.3). Define j = jγ j. Then one easily verifies the following result.

Lemma 3.12.1 (a) j is a 0-transversal of ζ : Γ˜ → Γ .

(b) Let χ : Γ → πC(Γ ) and χ : Γ → πC (Γ ) be the functions deﬁned 1 1 by χ(m)(jd1(m)) = d1j(m) and χ (m)(j d1(m)) = d1j (m) (m ∈ Γ) (see Eq.(3.1) in Sect. 3.4). Then

f χ (m) = χ(m) (m∈ Γ).

For a proﬁnite group G, denote by RC(G) the smallest closed normal subgroup of G such that G/RC(G) is a pro-C group. Clearly, RC(G) is the intersection of all closed normal subgroups N of G such that G/N is a pro-C group. (See RZ, Sect. 3.4, for properties of RC(G).)

Proposition 3.12.2 Let C and C be extension-closed pseudovarieties of ﬁnite groups, with C ⊆ C , and let Γ be a connected proﬁnite graph. Assume that

ζ : Γ˜ −→ Γ and ζ : Γ˜ −→ Γ 108 3 The Fundamental Group of a Proﬁnite Graph are a universal Galois C-covering and a universal Galois C -covering of Γ , respectively. Then

˜ = C \ ˜ C = C C Γ RC π1 (Γ ) Γ and π1 (Γ ) π1 (Γ )/RC π1 (Γ ) .

Proof We continue with the notation in the diagram above. By Proposition 3.1.5

C C ˜ =∼ C π1 (Γ )/π1 (Γ) π1 (Γ ), and hence

C ˜ ≥ C π1 (Γ) RC π1 (Γ ) . C C ˜ C = It follows that the pro- group π1 (Γ)/RC(π1 (Γ )) acts freely on C \ ˜ ˜ RC(π1 (Γ )) Γ , and the quotient graph of modulo this action coincides with Γ . Since Γ˜ is C-simply connected, the group

| ˜ = C ˜ C G( Γ) π1 (Γ)/RC π1 (Γ ) associated with the Galois C-covering → Γ˜ is trivial, and therefore

C ˜ = C π1 (Γ) RC π1 (Γ ) .

Thus the two assertions in the proposition follow.

Proposition 3.12.3 Assume that C is a pseudovariety of finite groups that is closed under extensions with abelian kernel, and let Ce be the smallest extension-closed pseudovariety of finite groups containing C. Let Γ be a connected profinite graph and let ζ : Γ˜ −→ Γ be its universal Galois C-covering. Then Γ˜ is a Ce-tree.

Proof By Corollary 3.10.5, this is equivalent to showing that the proﬁnite group Ce ˜ π1 (Γ) is perfect, i.e., it coincides with the closure of its commutator subgroup. Let ζ e : Γ˜ e → Γ be a universal Galois Ce-covering of Γ . By Proposition 3.1.5 there exists a Galois covering ψ : Γ˜ e → Γ˜ such that ζψ = ζ e. By Theorem 3.7.1 ψ Ce ˜ e| ˜ = Ce ˜ is a universal Galois -covering and G(Γ Γ) π1 (Γ). Consider the closure

= Ce ˜ Ce ˜ C π1 (Γ),π1 (Γ) Ce ˜ of the derived subgroup of π1 (Γ). Put

= C\Γ˜ e. 3.12 Change of Pseudovariety 109

The natural epimorphisms

β : Γ˜ e −→ and ϕ : −→ Γ˜ are Galois Ce-coverings with associated groups Ce ˜ C and π1 (Γ)/C, Ce ˜ respectively (see Proposition 3.6.1). According to Proposition 3.1.5, π1 (Γ) Ce Ce Ce π1 (Γ );soC π1 (Γ ). Hence the group π1 (Γ )/C acts freely on and

Ce \ = π1 (Γ )/C Γ.

Let ζ¯ e : → Γ be the corresponding Galois Ce-covering. Note that ζϕ= ζ¯ e.

ϕ β

α Γ˜ e Γ˜ ψ ζ¯ e ζ e ζ Γ

By Proposition 3.1.5 G(Γ˜ |Γ)= G(|Γ)/G(|Γ),˜ | ˜ | = C so that G( Γ)is an extension of G(Γ Γ) π1 (Γ ) with abelian kernel | ˜ = Ce ˜ G( Γ) π1 (Γ)/C. We observe that a ﬁnite abelian group belongs to a pseudovariety of ﬁnite groups that is closed under extensions with abelian kernel if and only if for every prime p that divides the order of this group, the cyclic group Cp of order p belongs to the e pseudovariety. Therefore, since Cp ∈ C if and only if Cp ∈ C , the group

| = Ce ˜ G( Γ) π1 (Γ)/C is a pro-C group. Therefore ζ¯ e is a Galois C-covering. Now, since ζ : Γ˜ → Γ is a universal Galois C-covering, there exists an epimorphism α : Γ˜ → such that ζ¯ eα = ζ . Then by Proposition 3.3.1 ϕα is an isomorphism, and therefore so is ϕ.It Ce ˜ follows that π1 (Γ)/Cis trivial, as required. Chapter 4 Proﬁnite Groups Acting on C-Trees

4.1 Fixed Points

Throughout this section π denotes a nonempty set of prime numbers.

Here we begin to study the structure of pro-π groups acting on π-trees (a pro- π group is an inverse limit of finite π-groups, in other words, finite groups whose orders are divisible only by primes in π). This section is concerned with fixed points. In particular, we shall prove that if a pro-π group acts on a π-tree, the subset of fixed points is a π-subtree (if it is not empty) and that a finite π-group acting on a π-tree fixes a vertex. If F = F(x,y) is a free pro-π group of rank 2, then the Cayley graph Γ = Γ(F,{x,y}) is a π-tree on which F acts (see Theorem 2.5.3); note that the quotient graph F \Γ is not a π-tree, but a bouquet of two loops (see Example 3.1.1). In contrast we have the following result.

Proposition 4.1.1 Let a pro-π group G act on a π-tree T . Let

N = Gm | m ∈ T , where Gm denotes the G-stabilizer of m. Then N\T is a π-tree.

Proof Let Cs(π) be the class of all ﬁnite solvable π-groups. By Proposition 3.10.4(b) T is Cs(π)-simply connected. Then, by Proposition 3.9.2,soisN\T . Therefore, by Corollary 3.10.2, N\T is a π-tree.

Theorem 4.1.2 Let a pro-π group G act freely on a π-tree T . Then G is a projective proﬁnite group.

Proof By Proposition 2.4.3(e), T is a p-tree for every p ∈ π, i.e.,

∗ d ε 0 −→ Zp E (T ), ∗ −→ Zp V(T) −→ Zp −→ 0

© Springer International Publishing AG 2017 111 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_4 112 4 Proﬁnite Groups Acting on C-Trees is a short exact sequence of free Zp-modules. Since the action of G on T is ∗ free, [[ Zp(E (T ), ∗)]] and [[ Zp(V (T ))]] are free [[ ZpG]] -modules (cf. RZ, Proposi- tion 5.7.1). Therefore the above sequence is a projective [[ ZpG]] -resolution of Zp. Hence cdp(G) ≤ 1, for each p ∈ π. Thus G is projective (see Sect. 1.11).

See Corollary 9.3.2(b) for an alternative proof of the above theorem using homology.

Corollary 4.1.3 Let a pro-π group G act on a π-tree T and let N = Gm | m ∈ T be the subgroup generated by the stabilizers of all m ∈ T . Then N is normal in G and G/N is projective.

Proof By Proposition 4.1.1, N\T is a π-tree. Clearly N is normal in G and G/N acts freely on N\T (see Lemma 2.2.1(b)). Then, by Theorem 4.1.2, G/N is projective.

Lemma 4.1.4 Let G be a finite group of prime order p generated by an element g. Let M be a free profinite [FpG]-module on a pointed profinite space (X, ∗) and let ν and τ be the endomorphisms of M defined by

− ν(m) = 1 + g +···+gp 1 m and τ(m)= (g − 1)m (m ∈ M), respectively. Then Im(ν) = Ker(τ).

Proof We first assume that the space (X, ∗) is finite. Since τ and ν commute with direct sums, it suffices to check in this case the result for a free module of rank 1, i.e., when M =[FpG]. Clearly τν = 0, and so Ker(τ) ≥ Im(ν). To prove equality, let m ∈[FpG] with τ(m)= 0. Say p−1 i m = aig (ai ∈ Fp). i=0 Then p−1 i τ(m)= (ai−1 − ai)g + ap−1 − a0 = 0. i=1 Therefore, a0 = a1 = ··· = ap−1, and so m = ν(a0). Thus, Ker(τ) ≤ Im(ν),as needed. Now, for a general profinite pointed space (X, ∗), write (X, ∗) = lim(X , ∗), ←− i where each (Xi, ∗) is a finite pointed space. Then M = lim M , ←− i where Mi is a free [FpG]-module with basis (Xi, ∗). Since ν and τ commute with inverse limits, the result now follows from the case above. 4.1 Fixed Points 113

Theorem 4.1.5 Suppose that a pro-π group G acts on a π-tree T . Then the following results hold. (a) The subset T G ={m ∈ T | gm = m, for all g ∈ G} of ﬁxed points of T under the action of G is either empty or a π-subtree of T . (b) If N is a closed normal subgroup of G and T N = ∅, then T N is a G-invariant π-subtree of T .

Proof Part (b) follows from (a). Here we prove part (a). If T G = T (i.e., the action is trivial), then the result is obvious. So we may suppose that the action of G on T is not trivial. From the continuity of the action we have that T G is a closed subset of T . Assume that T G = ∅. It is clear that T G is a proﬁnite subgraph of T with vertex set V(TG) = (V (T ))G, and that E∗(T G) = (E∗(T ))G. By Proposition 2.4.3(b), it is enough to check that T G is connected. We proceed in steps.

Step 1. Assume that G is cyclic of prime order p ∈ π. G By Proposition 2.3.2, it sufﬁces to check that H0(T , Fp) = 0. One may con- ∗ G G sider the modules [[ Fp((E (T )) , ∗)]] and [[ Fp(V (T )) ]] as submodules of ∗ [[ Fp(E (T ), ∗)]] and [[ FpV(T)]] , respectively. Then one has ∗ ∗ G ∗ ∗ G Fp E (T ), ∗ Fp E (T ) , ∗ = Fp E (T )/ E (T ) , ∗ and G G FpV(T) Fp V(T) = Fp V(T)/ V(T) , ∗ (see Lemma 2.4.6(c)). Since the only proper subgroup of G is the trivial group, it follows that the actions of G on the spaces

∗ ∗ E (T )/ E (T ) G, ∗ and V(T)/ V(T) G, ∗ are free, where ∗ is the image of (E∗(T ))G and of (V (T ))G, respectively. Hence the ∗ ∗ G G modules [[ Fp(E (T )/(E (T )) , ∗)]] and [[ Fp(V (T )/(V (T )) , ∗)]] are free, and therefore projective, [[ FpG]] -modules (cf. RZ, Propositions 5.7.1 and 5.4.2). Hence the natural [[ FpG]] -epimorphisms

∗ ∗ ∗ G Fp E (T ), ∗ −→ Fp E (T )/ E (T ) , ∗ and

G FpV(T) −→ Fp V(T)/ V(T) , ∗ split. So we can write the complex C(T,Fp) associated with T and Fp (see Sect. 2.3) in the following way:

∗ G d G ε 0 −→ Fp E (T ) , ∗ ⊕ M −→ Fp V(T) ⊕ L −→ Fp −→ 0, ∼ ∗ ∗ G ∼ G where M = [[ Fp(E (T )/(E (T )) , ∗)]] and L = [[ Fp(V (T )/(V (T )) , ∗)]] . 114 4 Proﬁnite Groups Acting on C-Trees

G We view C(T , Fp) as a subcomplex of the complex C(T,Fp) described above. G Since T is a p-tree, the sequence C(T,Fp) is exact. We note that T is connected if and only if

∗ G ∗ = G d Fp E (T ) , Ker(ε|[[Fp(V (T )) ]] ). To prove this equality observe ﬁrst that clearly

∗ G ∗ ≤ G d Fp E (T ) , Ker(ε|[[Fp(V (T )) ]] ).

∈ G ∈ Next take b Ker(ε|[[Fp(V (T )) ]] ). Since C(T,Fp) is exact, there exists an a ∗ ∗ G [[ Fp(E (T ), ∗)]] with d(a) = b.Leta = a0 + a1 for some a0 ∈[[Fp((E (T )) , ∗)]] and a1 ∈ M. Then d(a1) = b − d(a0) is an element of d(M) whichisfixedbyG, since d(a0) and b arefixedbyG.Letg be a generator of G and define endomorphisms ν and τ of [[ Zp(V (T ))]] by

p−1 ν(m) = 1 + g +···+g m and τ(m)= (g − 1)m m ∈ Zp V(T) . Then, according to Lemma 4.1.4, the image of ν coincides with the kernel of τ on any free [ZpG]-submodule of [[ Fp(V (T ))]] . Since d(M) is a free [[ FpG]] -module (because d in an injection), and since d(a1) ∈ Ker(τ) (because d(a1) is ﬁxed by G), one has that d(a1) = ν(d(a2)),forsomea2 ∈ M. Write

G d(a2) = b0 + b1, for some b0 ∈ Fp V(T) and b1 ∈ L.

Then d(a1) = ν(b0) + ν(b1). Now, the element

ν(b1) = d(a1) − ν(b0) = b − d(a0) − ν(b0) G G lies in L ∩[[Fp(V (T ) )]] , since ν(b1) ∈ L and b,d(a0), ν(b0) ∈[[Fp(V (T ) )]] . Therefore, ν(b1) = 0. Hence

d(a1) = ν(b0) = pb0 = 0. ∗ G It follows that b = d(a0), and thus, b ∈ d([[ Fp((E (T )) , ∗)]] ).

Step 2. Let G =∼ Z/nZ, where n is a natural number which is divisible only by primes in π. Assume that T G is not connected. Consider a minimal such n with T G not connected. By Step 1, n is not prime. Pick a proper nontrivial subgroup H of G. Since |H|

Step 3. Assume now that G is procyclic. For a natural number m, put Gm ={gm | g ∈ G}; then G/Gm is a ﬁnite cyclic π- group. Since T = T G, there exists some vertex of T with a nontrivial stabilizer, say n n n G 0 , where n0 is a natural number. For n0|n,wehaveG ≤ G 0 . Then = n G lim←− G/G . n0|n 4.1 Fixed Points 115

n By Proposition 4.1.1, the quotient graph Tn = G \T is a π-tree for any n which is a multiple of n0. Clearly = T lim←− Tn. n0|n n n G/G Furthermore, G/G acts on Tn in a natural way. By Step 2, Tn is a π-tree. Therefore, by Proposition 2.4.3(d), G = G/Gn T lim←− Tn n0|n is also a π-tree.

Step 4. Let G be an arbitrary pro-π group. Obviously, T G = T g. g∈G

By Step 3, T g is a π-subtree of T for every g ∈ G. Hence, by Proposition 2.4.9, T G is a π-tree.

Corollary 4.1.6 Suppose that a pro-π group G acts on a π-tree T , and let v and w be two different vertices of T . Then the set of edges E([v,w]) of the chain [v,w] is nonempty, and Gv ∩ Gw ≤ Ge for every e ∈ E([v,w]).

Proof Clearly E([v,w]) = ∅, because otherwise [v,w]=V([v,w]), which is not K connected (see Lemma 2.1.9). Let K = Gv ∩ Gw. Since v,w ∈ T , we have that T K is not empty, and hence, by Theorem 4.1.5, T K is a π-tree. So [v,w]⊆T K (see Lemma 2.4.10). Therefore K ≤ Ge, for every edge e ∈[v,w].

Corollary 4.1.7 Suppose that a pro-π group G acts on a π-tree T . If T contains two G-invariant π-subtrees T1 and T2 which are disjoint, then G (a) there exists an edge e ∈ T such that e/∈ T1 and e/∈ T2; G = ∅ = G (b) T1 T2 ; (c) if G acts freely on a subtree T1 of T , then it acts freely on T .

Proof (a) Consider the profinite graph T obtained from T by collapsing T1 to a point v1 and T2 to a different point v2. By Lemma 2.4.7, T is a π-tree. Moreover, G G acts on T fixing v1 and v2. By Theorem 4.1.5, T is a π-tree; and by Corol- G G lary 4.1.6, the chain [v1,v2] in T contains an edge e. Therefore e ∈ T with e/∈ T1 and e/∈ T2 (by abuse of notation, we denote the edge e and its unique lifting in T by the same letter). In particular, T G is not empty, and so it is a π-tree. G (b) It follows that T1 ∩ T = ∅ (for otherwise, according to (a), there would exist some fixed edge of T not in T G, which is obviously not the case), and in particular G = ∅ G = ∅ T1 . Similarly T2 . 116 4 Profinite Groups Acting on C-Trees

(c) Say G acts freely on T1.Letv be a vertex of T such that its stabilizer Gv is ={ } Gv = ∅ not trivial. Put T2 v . Then T1 and T2 are disjoint. So, according to (b), T1 , a contradiction. Hence G acts freely on T , as asserted.

Next we prove that a ﬁnite π-group G (i.e., if a prime number p divides the order of G, then p ∈ π) acting on a π-tree must ﬁx a vertex.

Theorem 4.1.8 Suppose that a ﬁnite π-group G acts on a π-tree T . Then G = Gv, for some vertex v ∈ V(T).

Proof We must prove that T G ={t ∈ T | gt = t,∀g ∈ G}=∅. We use induction on the order of G. Assume that |G| > 1 and that the result holds whenever a finite π-group L acts on a π-tree and |L| < |G|. Observe first that if G has prime order and we had T G =∅, then the action of G on T would be free; hence Theorem 4.1.2 would imply that G is projective, contradicting the fact that G is finite and nontrivial; so if G has prime order, the result follows. If G is not simple and N is a proper nontrivial normal subgroup of G, the induction hypothesis implies that T N = ∅. Hence by Theorem 4.1.5, T N is a π-subtree of T . Furthermore, G/N acts naturally on T N ; hence by induction again, one has that T G = (T N )G/N = ∅. Therefore if G is not simple, the result follows. Let us consider the remaining case: suppose that G is a nonabelian simple group. G Assume that T =∅. Choose a minimal G-invariant π-subtree T0 of T (see Propo- sition 2.4.12(a)). Fix a maximal proper subgroup H of G. Then the set − J = x ∈ G − H xHx 1 ∩ H = 1 is not empty, since otherwise G would be a Frobenius group and hence not simple H = ∅ (cf. Huppert 1967, Chap. V, Theorem 7.6). By the induction hypothesis, T0 ;let ∈ H = = v V(T0 ). We note that H Gv,theG-stabilizer of v, that is, xv v, for every x ∈ G − H , because if xv = v for some x ∈ G − H , we would have that G = H,x fixes v, contradicting the assumption that T G =∅. Choose g ∈ J such that the chain P =[v,gv] is minimal in the set of chains {[v,xv]|x ∈ J }, ordered by inclusion. Observe that since v,gv ∈ T0, we have that P =[v,gv]⊆T0.

Claim A For every h ∈ H , Wh ={w ∈ V(P)| ghw = w}=∅and v,gv∈ / Wh.

The second statement is clear. To prove the first one consider the following subgraph of T0 n−1 D = (gh)iP, i=0 where n is the order of gh. Since h fixes v, one has that + (gh)igv ∈ (gh)iP ∩ (gh)i 1P, for i = 0,...,n− 1; 4.1 Fixed Points 117 hence D is connected (see Lemma 2.1.7(b)), and therefore a π-subtree of T0 (see Proposition 2.4.3(b)). Since gh is a proper subgroup of G and acts on D, there exists a vertex w of D fixed by gh. Note that w ∈ P ; indeed, say w ∈ (gh)iP ; then w = (gh)−iw ∈ P .ThisprovesClaimA.

Claim B For h ∈ H and every w ∈ Wh one has [v,w]=P = [w,gv].

We observe ﬁrst that the inequalities [v,w]=P = [w,gv] are equivalent if w ∈ Wh. Indeed, let us suppose that [v,w]=P ; then ghP = gh[v,w]=[gv,w]⊆P ; therefore, P = (gh)nP ⊆ (gh)n−1P ⊆···⊆ghP =[gv,w]⊆P ; thus [gv,w]=P . Similarly, if [w,gv]=P , one sees that [v,w]=P . ˜ ∈ ∈ Next we prove that there exists at least one h H and some wh˜ Wh˜ such that [ ]= ∈ [ ]= v,wh˜ P . Assume on the contrary that for every h H , one has v,w P (and hence [w,gv]=P ), for all w ∈ Wh. Then ghP = P , for all h ∈ H . Since G is generated by the set {gh | h ∈ H}, it follows that P is G-invariant. Set K = gHg−1 ∩H ; then, by the choice of g and v, K = 1 and v,gv ∈ T K . By Theorem 4.1.5, T K is a π-tree; therefore P ⊆ T K , i.e., K acts trivially on P . Since G is simple nonabelian, there is a maximal proper subgroup M of G with K ≤ M. Then G = K,M,so that P G = P M . By the induction hypothesis, P M = ∅, and this contradicts the assumption that T G =∅. ˜ ∈ ∈ [ ]= Therefore, there exist some h H and a vertex wh˜ Wh˜ such that v,wh˜ P [ ]= ∈ (and hence wh˜ ,gv P ). To complete the proof of Claim B,leth H , and let ∈ ∈[ ] ∈[ ] ∈[ ] [ ]⊆ w Wh. Clearly either w v,wh˜ or w wh˜ ,gv ;sayw v,wh˜ . Then v,w [ ] [ ]= v,wh˜ . Thus v,w P .ThisprovesClaimB. For each h ∈ H , choose a vertex v ∈ W . Set h h D1 = [v,vh],D2 = [vh,gv] and D0 = D1 ∩ D2. h∈H h∈H

Note that D1,D2,D0 ⊆ P ; furthermore, v/∈ D2 and gv∈ / D1 by Claim B.The proﬁnite subgraphs D1 and D2 are nonempty and connected (see Lemma 2.1.7); therefore, by Proposition 2.4.3(b), they are π-subtrees of T0. Since D0 contains all the vertices vh (h ∈ H), it is not empty. It follows from Proposition 2.4.9 that it is a ⊆ π-subtree of T0. Since T0 is G-invariant, one has that x∈G xD0 T0. = Claim C T0 x∈G xD0.

Since T0 is a minimal G-invariant subtree of T , we just need to show that x∈G xD0 is a G-invariant π-subtree of T0. Since G is finite, x∈G xD0 is closed in T0, and obviously it is G-invariant. Since G = gh| h ∈ H and since gh fixes ∈ avertexinD0 for every h H , we deduce that x∈G xD0 is connected (see Lemma 2.2.4). Then, by Proposition 2.4.3(b), x∈G xD0 is a π-subtree. This proves Claim C. 118 4 Profinite Groups Acting on C-Trees

Since v ∈ T0, it follows from Claim C that there exists a g0 ∈ G such that g0v ∈ D0 = D1 ∩ D2. Next we show that g0 ∈ J . Since v/∈ D2, we deduce that g0v = v, −1 hence g0 ∈/ H . As remarked in the proof of Claim B, the subgroup K = gHg ∩ H −1 acts trivially on P . Therefore, K is contained in the stabilizer g0Hg0 of the vertex g0v ∈ D0 ⊂ P . Since = ⊆ −1 ∩ 1 K g0Hg0 H, we have that g0 ∈ J . To ﬁnish the proof we ﬁrst note that the chain [v,g0v] is contained in D1 because v,g0v ∈ D1. Since gv∈ / D1, we have that [v,g0v] is a proper subset of P =[v,gv]. This contradicts the minimality of [v,gv]. Therefore T G = ∅, as desired.

Corollary 4.1.9 Let G be a pro-π group acting on a π-tree T and let D be a G- invariant π-subtree of T . Then for every vertex v in T , there exists a vertex w ∈ D such that Gv ≤ Gw. In particular, if the action of G on D is free, then so is the action of G on T (see also Corollary 4.1.7(c)).

Proof Write = Gv lim←− Gv/N, N∈N where N is the collection of all open normal subgroups of Gv. For each N ∈ N , set TN = N\T . Consider the natural actions of Gv and Gv/N on TN . Since N is generated by the N-stabilizers of the vertices of T , it follows from Proposition 4.1.1 that each TN is a π-tree. Let DN be the image of D in TN under the projection T −→ TN . Since D is connected, so is DN ; hence DN is a π-tree on which the ﬁnite π-group Gv/N acts. G G /N G Since (DN ) v = (DN ) v , we have that (DN ) v = ∅, by Theorem 4.1.8. Observe that the collection {DN | N ∈ N } is naturally an inverse system, and = D lim←− DN . N∈N G We easily deduce that the corresponding collection {(DN ) v | N ∈ N } of sets of ﬁxed points is an inverse system, and Gv = Gv D lim←− (DN ) . N∈N G G Therefore D v = ∅ (see Sect. 1.1). For every vertex w ∈ D v we have Gv ≤ Gw.

Using this corollary we can now complete the proof of Proposition 2.4.12 on the uniqueness of minimal G-invariant subtrees of a G-tree, in certain cases.

Proof of Proposition 2.4.12(b) Assume that |D| > 1, and let D0 be another minimal G-invariant π-subtree of T . Denote by T the proﬁnite graph obtained from T by collapsing D0 to a point (see Example 2.1.2). Since D0 is G-invariant, the action of G carries over naturally to T . By Lemma 2.4.7, T is a π-tree. Since D and D0 4.2 Faithful and Irreducible Actions 119 are minimal G-invariant, D ∩ D0 =∅. This means that the π-tree D is isomorphic to its image D in T and, in particular, |D | > 1. Observe that D is a minimal

G-invariant π-subtree of T .Letv be the vertex of T to which D0 collapses. Then

G = Gv. By Corollary 4.1.9, there is a vertex w in V(D) such that Gv ≤ Gw; hence

Gw = G, and so G ﬁxes the vertex w. But since D is minimal G-invariant, we have D ={w}, a contradiction.

4.2 Faithful and Irreducible Actions

We say that an action of a pro-π group G on a π-tree T is irreducible (or that G acts irreducibly on T )ifT is a minimal G-invariant π-subtree of T . We say that the action is faithful (or that G acts faithfully on T ) if the kernel CG(T ) of the action G −→ Aut(T ) is trivial, i.e., if whenever g ∈ G and gm = m for all m ∈ T , then g = 1.

Remark 4.2.1 (a) If G acts faithfully and irreducibly on T , then either G is trivial (in which case |T |=1) or |V(T)| > 1. Consequently (see Theorem 4.1.8), if in addition G is ﬁnite, it must be trivial. (b) Let H be a closed subgroup of a pro-π group G, and assume that G acts on a π-tree T . Observe that if the action of G on T is faithful, so is the induced action of H on T ; and if the induced action of H on T is irreducible, so is the action of G on T . (c) If a pro-π group G acts freely on a π-tree T , then this action is faithful; moreover it is obvious that such a group acts faithfully and irreducibly on any minimal G-invariant π-subtree of T .

An often essential part of the information carried by groups acting on profinite trees can be obtained through faithful and irreducible actions. Indeed, if a profinite group G acts on a profinite tree T , one can consider the action of G on a minimal G-invariant subtree D and factor out the kernel of this action CG(D). The resulting action of G/CG(D) on D is faithful and irreducible. In this section we study faithful and irreducible actions; this will lead us eventually to a classification of pro-π groups acting on π-trees (Theorem 4.2.11).

Proposition 4.2.2 Let G beapro-π group acting irreducibly on a π-tree T , and let N be a closed normal subgroup of G. (a) If N is contained in the stabilizer of some vertex, then N acts trivially on T . (b) If N is a ﬁnite normal subgroup of G, then N acts trivially on T .

Proof Part (b) follows from part (a) and Theorem 4.1.8. To prove part (a) note ﬁrst that since T N is not empty, it is a G-invariant π-subtree of T (see Theo- rem 4.1.5(b)). The irreducibility of the action of G implies that T N = T . 120 4 Proﬁnite Groups Acting on C-Trees

Recall that a closed subgroup N of a proﬁnite group G is called subnormal if there exists a ﬁnite chain of closed subgroups of G,

N = N0 N1 ···Nk = G, each normal in the next one. We refer to this series as a normal series joining N to G.

Proposition 4.2.3 Let G be a pro-π group acting faithfully and irreducibly on a π-tree T . Then the following statements hold. (a) Every closed subgroup of G that contains a nontrivial closed subnormal subgroup of G acts irreducibly on T . (b) If G is abelian, then it acts freely on T . Consequently, either G is trivial or ∼ G = Zρˆ , for some ρ ⊆ π.

Proof If |T |=1, then |G|=1 and the results are obvious. Assume |T | > 1. (a) It sufﬁces to prove that every nontrivial closed subnormal subgroup of G acts irreducibly on T (see Remark 4.2.1(b)). For a closed subnormal subgroup N of G, let k = kN denote the smallest natural number such that

N = N0 N1 ···Nk = G is a normal series joining N to G. We use induction on k. Let N be a nontrivial closed subnormal subgroup of G and assume that kN = 1, i.e., N is normal in G. Choose a minimal N-invariant π-subtree D of T (see Propo- sition 2.4.12(a)). We claim that |D| > 1. Indeed, if D ={v} for some v ∈ V(T), then N ≤ Gv. By Proposition 4.2.2, N must act trivially on the whole of T , contradicting the faithfulness of the action of G on T . Thus |D| > 1. Since N is normal in G, the shift gD of D by any element g of G is also a minimal N-invariant π-subtree of T . By Proposition 2.4.12(b), such a subtree is unique, so that gD = D for all g ∈ G. Since G acts irreducibly on T , one concludes that D = T , proving that N acts irreducibly on T . Let k = kN > 1 and assume that the result holds whenever M is a closed subnormal subgroup of G with kM

The next result gives a description of direct products acting faithfully and irreducibly on a proﬁnite tree. 4.2 Faithful and Irreducible Actions 121

Proposition 4.2.4 Assume that a pro-π group G acts faithfully and irreducibly on a π-tree T , and assume that G can be written as a direct product G = G1 × G2 of nontrivial closed subgroups G1 and G2. Then G is a projective proﬁnite group and its action on T is free. Furthermore, G1 and G2 have relatively prime orders.

Proof Let (1, 1) = (g1,g2) ∈ G1 × G2;sayg1 = 1 for deﬁniteness. Since G2 G and g1×G2 ≥ G2, we deduce from Proposition 4.2.3(a) that g1×G2 acts faithfully and irreducibly on T ; similarly, since

g1× g2≥ g1 and g1 g1×G2, so does g1× g2; and again, since (g1,g2) g1× g2, we get that (g1,g2) acts irreducibly on T . This means in particular that (g1,g2)t = t, for every t ∈ T . Since (g1,g2) was chosen arbitrarily in G, it follows that G acts freely on T . Then, by Theorem 4.1.2, G is projective. If the orders of G1 and G2 hadaprimep in common, then G would have a subgroup isomorphic to Zp × Zp. Since cdp(Zp × Zp) = 2 (cf. RZ, Exercise 7.4.3), we would have that cdp(G) ≥ 2, and hence G would not be projective. Thus G1 and G2 have relatively prime orders.

Our next aim is to prove a structure theorem on pro-π groups acting faithfully and irreducibly on π-trees. We do this in Theorem 4.2.10. First we need a series of auxiliary results.

Lemma 4.2.5 Let G be a proﬁnite group whose p-Sylow subgroups are procyclic for every prime number p (a proﬁnite Zassenhaus group). Then G = M N, where M and N are procyclic groups of relatively prime order.

Proof This follows easily by a standard inverse limit argument from the corresponding property for finite groups (Zassenhaus groups). We just sketch this argument. Write G as the inverse limit of all its finite quotient groups GU = G/U , where U ranges through the set U of all the open normal subgroups U of G.ForU ∈ U,let SU be the set of all pairs (K, H ), where K and H are cyclic subgroups of GU of relatively prime order, K GU and GU = KH. Then SU = ∅ (cf. Hall 1959, Theo- rem 9.4.3). If V,U ∈ U and V ≤ U, the natural epimorphism of groups GV → GU induces a map of sets ϕVU : SV → SU . One checks that (SU ,ϕVU) is an inverse system of finite nonempty sets; hence its limit is not empty (see Sect. 1.1). Let {(M ,N )} ∈U ∈ lim S . Define M = lim M and N = lim N . Then M and N U U U ←− U ←− U ←− U are procyclic subgroups of G of relatively prime order, M G and G = MN = M N, as needed.

We recall that a profinite group G is called Frobenius if it contains a nontrivial closed Hall subgroup H (i.e., the profinite order #H of H and its index [G : H ] in G are relatively prime) with H ∩ H g = 1 for all g ∈ G − H (cf. Sect. 4.6 in RZ). If σ is a set of prime numbers with 2 ∈ σ , then the infinite dihedral pro-σ group is the 122 4 Profinite Groups Acting on C-Trees

∼ −1 semidirect product α b, where α = Zσˆ , the order of b is 2, and bαb = α . Put a = αb; then a2 = 1. One checks easily that α b= a b, the free pro-σ product of a and b. In particular, if σ ={2}, we get the inﬁnite dihedral pro-2 group ∼ Z2 C2 = C2 C2, the free pro-2 product of two copies of C2 (the action of C2 on Z2 is by inversion).

Lemma 4.2.6 Let G be a pro-π group acting faithfully and irreducibly on a π-tree T and let A be a nontrivial abelian closed normal subgroup of G. Then: ∼ (a) A is an inﬁnite procyclic group A = Zσˆ , for some subset of primes σ of π. (b) Let H be a closed subgroup of G such that the induced action of H on T is reducible. Then (b1) hah−1 = a, for all 1 = a ∈ A and 1 = h ∈ H ; and (b2) if, in addition, p ∈ σ and p does not divide the order of H , then H is a ﬁnite cyclic group isomorphic with a subgroup of Cp−1.

(c) The centralizer CG(A) of A in G is a projective profinite group, and it acts freely on T . (d) If H is a nontrivial finite subgroup of G, then the induced action of H on T is reducible. Moreover, ∼ (1) if 2 ∈/ σ , then H = Cn, for some natural number n and ∼ A,H=A H(= Zσˆ Cn) is a profinite Frobenius group with isolated subgroup H . Furthermore, n divides p − 1, for all p ∈ σ ; and ∼ (2) if 2 ∈ σ , then H = C2 and ∼ A,H=A H(= Zσˆ C2) is an infinite dihedral pro-σ group. (e) If G has elements of order 2, and B is the closure of the subgroup of G generated by all elements of order 2, then B = CB (A) b, where b ∈ B is any element of order 2.

Proof (a) This follows from Proposition 4.2.3. (b1) By contradiction. Suppose that hah−1 = a,forsome1= h ∈ H and 1 = a ∈ A. Then h h, a≥ aA G. Applying Proposition 4.2.3(a), one gets that h acts irreducibly on T , and hence (see Remark 4.2.1(b)) so does H , a contradiction. 4.2 Faithful and Irreducible Actions 123 ∼ (b2) By (a), A = Zσˆ and so the p-Sylow subgroup Ap of A is isomorphic to Zp. Since Ap is characteristic in G,themap

ϕp : H −→ Aut(Ap), which associates with h ∈ H the restriction of the inner automorphisms of G determined by h, is a homomorphism. Note that ϕp is injective by (b1). Now (cf. RZ, ∼ Theorem 4.4.7), if p = 2, then Aut(Ap) = Zp × Cp−1, so that ϕp(H ) is isomorphic to a subgroup of Cp− , because p does not divide the order of H .Ifp = 2, ∼ 1 Aut(A2) = Z2 × C2; therefore, H = 1. (c) By part (b1), CG(A) ∩ Gv = 1, for all v ∈ V(T); and this means that CG(A) acts freely on T . Then, by Theorem 4.1.2,CG(A) is projective. (d) Since H is ﬁnite and A is torsion-free and normal, one has that A,H=A H.

Also,wehavethatH = Hv,forsomev ∈ V(T), by Theorem 4.1.8. Since H = 1, then G = 1, and so |V(T)| > 1. Hence the action of H on T is reducible. It follows from (b1) that the natural homomorphism

ϕp : H −→ Aut(Ap) ∼ is injective for every p ∈ σ .If2∈ σ , this implies that H = C2, because H is finite; therefore ∼ A,H=A H = Zσˆ C2. 2 ∼ It follows from (b1) that if H = h (with h = 1), the action of h on A = Zσˆ is by inversion: ah = a−1 (a ∈ A); i.e., A,H is the infinite dihedral pro-σ group. On the other hand, if 2 ∈/ σ , then H is isomorphic to a subgroup of Cp− ,for ∼ 1 every p ∈ σ . In particular, H is a cyclic group, say H = Cn. By (b1), the action of H on A is fixed-point-free. Since we also have that the orders of H and A are relatively prime, this means that A,H=A H is a profinite Frobenius group with isolated subgroup H (cf. RZ, Theorem 4.6.9 (d)). (e) Let b1 and b2 be elements of order 2 in B, and let a be any element of A. −1 By (d), one has b1ab1 = b2ab2 = a , so that b1b2 ∈ CB (A).Let = ∈ 2 = 2 = K b1b2 b1,b2 B,b1 b2 1 .

Then K is normal in B and K ≤ CB (A) ≤ B; moreover, by (c), CB (A) is torsion- free. Clearly B/K has order 2. Therefore, K = CB (A). Thus, B = CB (A) b,for any element b of B of order 2.

In the next result we shall use the following notation. Let G be a pro-π group that acts on a π-tree T .IfU is a closed normal subgroup of G, we denote by U˜ the closed subgroup of U generated by the U-stabilizers of the vertices v of T : ˜ U = Uv v ∈ V(T). ˜ −1 Note that U is a normal subgroup of G,forifx ∈ Uv and g ∈ G, then gxg ∈ Ggv ∩ U = Ugv (v ∈ V(T)). 124 4 Proﬁnite Groups Acting on C-Trees

Lemma 4.2.7 Let π be a nonempty set of primes. Let G be a nontrivial pro-π group that acts faithfully and irreducibly on a π-tree T . For an open normal subgroup U, ˜ ˜ put GU = G/U, where U is as above. Then there exists a family W ={W(U) | U ∈ U} of open normal subgroups of G such that G can be represented as an inverse limit = G lim←− GW(U) U∈U of pro-π groups, where, for each W = W(U), the group GW acts faithfully and irreducibly on a π-tree DW with ﬁnite stabilizers of vertices, and such that the open ˜ normal subgroup AW = W/W of GW acts freely on DW . Moreover, one has the following results.

(a) If G\T is ﬁnite, so is GW \DW , for every W ∈ W. (b) Let C be an extension-closed pseudovariety of ﬁnite groups such that π(C) = π. If T is C-simply connected, so is DW , for each W ∈ W. (c) If G has no nonabelian pro-p subgroups with induced free action on T , then (c1) for each W ∈ W, ˜ ∼ AW = W/W = A B = Zσˆ Zρˆ, ∼ ∼ where 1 = A = Zσˆ , B = Zρˆ , σ and ρ are disjoint sets of prime numbers; consequently, A is normal in GW ; ˜ (c2) if G is a pro-p group, then each AW = W/W is isomorphic with Zp; (c3) GW does not contain any nonabelian free pro-p group, for any p ∈ π.

Proof Let U be the collection of all open normal subgroups of G.ForU ∈ U,set ˜ ˜ TU = U\T . By Proposition 4.1.1, TU is a π-tree; furthermore, U/U acts freely on TU .Ifm ∈ T , then Um = Gm ∩ U, for every U ∈ U. Hence the action of GU on TU has ﬁnite vertex stabilizers. ˜ ∩ ˜ ≥ ∩ ∈ U ˜ = { | ∈ U} Note that U V U V ,forU,V and U∈U U 1; so GU U and {TU | U ∈ U} are inverse systems over U, and = = G lim←− GU and T lim←− TU . (4.1) U∈U U∈U

Consider the natural action of GU on TU . Since G = 1, we have |V(T)| > 1 and T G =∅. Since G = GU =∅ T lim←− TU , U∈U

GV =∅ there exists an open normal subgroup V of G such that TV (see Sect. 1.1). GU ≤ ∈ U Thus, TU is empty for all U V , U . Then, by Proposition 2.4.12, there exists a unique minimal GU -invariant π-subtree DU of TU , whenever U ∈ U with U ≤ V . Clearly, for such U, GU acts irreducibly on DU . Denote by KU the kernel of the action of GU on DU , i.e., KU consists of those elements x ∈ GU such that xd = d, for all d ∈ DU . 4.2 Faithful and Irreducible Actions 125

Next we shall deﬁne a subcollection W of open normal subgroups of G in such a way that in addition, for each W ∈ W, the action of GW on DW is faithful and equalities analogous to (4.1) hold. Let

ϕU : G −→ GU (U ∈ U,U ≤ V) and : −→ : −→ ≤ ≤ ∈ U ϕU1,U2 GU1 GU2 ,ψU1,U2 TU1 TU2 (U1 U2 V,U1,U2 ) denote the canonical epimorphisms. Note that for U1 ≤ U2 ≤ V (U1,U2 ∈ U), one has ⊆ DU2 ψU1,U2 (DU1 ), because DU2 is the unique minimal GU2 -invariant subtree of TU2 . This implies that ≤ ϕU1,U2 (KU1 ) KU2 . Thus, the groups KU form an inverse system, and so do the groups GU /KU (U ∈ U,U ≤ V ). For U ∈ U, U ≤ V , deﬁne = = −1 W W(U) UϕU (KU ).

Clearly W ∈ U. By deﬁnition GW acts irreducibly on DW . We claim that this action ˜ is also faithful or, equivalently, that KW = 1. Note ﬁrst that ϕU (W) = (U/U)KU . ˜ Now, for d ∈ DU , we have that (ϕU (W))d = KU , since U/U acts freely on TU ; hence, by Corollary 4.1.9, (ϕU (W))v ≤ KU , for all v ∈ TU . Therefore the subgroup of ϕU (W) generated by the ϕU (W)-stabilizers of the vertices of TU is KU .So(see ˜ = ˜ = −1 ˜ ≥ ˜ Lemma 2.2.1(b)), ϕU (W) KU . We deduce that W ϕU (KU ), since W U. Thus, ˜ ∼ GW = G/W = GU /KU . On the other hand, ˜ ∼ TW = W\T = KU \TU , and these isomorphisms are compatible with the actions of GW on TW and of GU /KU on KU \TU . Now, the unique GU /KU -invariant minimal π-subtree of KU \TU is KU \DU = DU , since KU acts trivially on DU . Obviously GU /KU acts faithfully on DU , and thus the claim is proved. Let W = = −1 ∈ U ≤ W(U) UϕU (KU ) U ,U V . Observe that

{GW(U) | U ∈ U} and {GU /KU | U ∈ U} are isomorphic inverse systems so that = lim←− GW(U) lim←− GU /KU . U∈U U∈U 126 4 Proﬁnite Groups Acting on C-Trees

So it remains to prove that = lim←− GU /KU G. U∈U From the exactness of

1 −→ KU −→ GU −→ GU /KU −→ 1 and the fact that the functor lim is exact (cf. RZ, Proposition 2.2.4), it follows that ←− this is equivalent to showing that = lim←− KU 1. U∈U To show this, suppose g ∈ lim K . To prove that g = 1, it sufﬁces to see that g ∈ G ←− U v for all v ∈ V(T), because this would mean that g is in the kernel of the action of G on T (which is trivial, since the action of G on T is faithful). Let

ψU : T −→ TU be the natural epimorphism. Put gU = ϕU (g), vU = ψU (v) (U ∈ U). To prove that ∈ ∈ ∈ U g Gv one just needs to show that gU (GU )vU for any U.ForU ,let U = ψU ,U (DU ).

U oG U ≤U

Clearly, gU ∈ ϕU ,U (KU ) ≤ KU for every open normal subgroup U of G contained in U, and therefore gU fixes all points of U . Hence it is enough to prove that U = TU . One can easily see that for a pair of open normal subgroups U1,U2 of G with U1 ≤ U2 ≤ U, the inclusion ψU ,U (U ) ⊆ U holds. This means that the 1 2 1 2 sets in the collection {U | U o G, U ≤ U} form an inverse system. Since U is G -invariant for every U , lim( ) is a G-invariant π-subtree of T . There- U ←− U fore lim( ) = T , because the action of G on T is irreducible. This implies that ←− U = T , as desired. Thus, lim K = 1, and so we deduce that lim G /K = G. U U ←− U ←− U U This completes the proof of the first part of the lemma. The proof of (a) follows from the fact that GW \DW ⊆ GW \TW , and GW \TW is an epimorphic image of G\T . To see (b) note first that, according to Proposi- tion 3.9.2, since T is C-simply connected, so is TU . Then use Proposition 3.7.3(d) to get that DU is C-simply connected. Next we prove parts (c1), (c2) and (c3). By Corollary 4.1.3 W/W˜ is a projective profinite group. Therefore there exists a closed subgroup H of W such that the restriction (ϕW )|H of the natural homomorphism ϕW to H is an isomorphism onto ˜ W/W .Letv ∈ V(T)and consider the image ψW (v) of v in TW . Then ∩ ≤ ˜ ∩ = ϕW (H Gv) (W/W) (GW )ψW (v) 1, ∩ because (GW )ψW (v) is finite as shown above. This means that H Gv is trivial for all v ∈ V(T); therefore H acts freely on T . It then follows from our hypothesis 4.2 Faithful and Irreducible Actions 127 that H has no nonabelian pro-p subgroups. Thus all the Sylow subgroups of H , and therefore of W/W˜ , are procyclic. Since W/W˜ is projective, it is torsion-free; it follows that ˜ ∼ W/W = Zσˆ Zρˆ, where σ and ρ are disjoint sets of prime numbers (see Lemma 4.2.5). Therefore ˜ ∼ W/W contains a characteristic subgroup K such that K = Zσˆ . Hence K is normal in GW . This proves part (c1). To prove part (c2) we observe that if G is a pro-p group, then so is W/W˜ . There- ˜ ∼ fore, by part (c1), W/W = Zp. Finally, we prove (c3). Suppose LW is a nonabelian free pro-p subgroup of GW . ≤ : −→ Let L G be such that ϕW |L L LW is an isomorphism (such an L exists since LW is a projective profinite group). Since AW is open in GW and it acts freely on TW ,theL-stabilizers of vertices of TW are finite, and hence trivial; i.e., LW acts freely on TW . Since ϕW sends the stabilizer Lv of a vertex v ∈ T isomorphically to the LW -stabilizer of ψ(v)∈ TW , we deduce that L acts freely on T , a contradiction.

We can now prove a structural result for pro-p groups that act faithfully and irreducibly on a π-tree.

Proposition 4.2.8 Let p ∈ π, where π is a set of prime numbers. Let G be a nontrivial pro-p group that acts faithfully and irreducibly on a π-tree T . Then one of the following assertions holds: (a) G has a nonabelian closed subgroup whose induced action on T is free (such a subgroup is necessarily a free pro-p group of rank at least 2). ∼ (b) G = Zp. ∼ ∼ ∼ (c) G = Z C(= Z2 C2), where Z = Z2, C = C2 and if C = g, then g acts on Z by inversion; i.e., G is an inﬁnite dihedral pro-2 group.

Proof If G has a nonabelian closed subgroup H acting freely on T , then H is a projective pro-p group by Theorem 4.1.2, and therefore it is a free pro-p group (see Sect. 1.11). Suppose that G does not satisfy (a). By Lemma 4.2.7(c2), G admits a decomposition G = lim G , ←− W where W ranges through a certain collection W of open subgroups of G, each GW is a quotient group of G that acts faithfully and irreducibly on a π-tree DW with finite stabilizers of vertices and so that each GW has an open normal subgroup AW isomorphic with Zp. Assume that p>2. Consider any vertex v of DW ; then, since (GW )v is a finite p-group, and since its order divides p − 1 (see Lemma 4.2.6(d)), we have that (GW )v = 1. Thus, GW acts freely on DW . By Theorem 4.1.2, GW is a projective 128 4 Profinite Groups Acting on C-Trees group; and since GW is pro-p, it is a free pro-p group (see Sect. 1.11). Since AW is an open subgroup of G and A =∼ Z , one deduces that rank(G ) = 1(cf. W ∼ W p W RZ, Theorem 3.6.2(b)), i.e., GW = Zp. Since this is the case for every W and since every surjective endomorphism of Zp is an isomorphism, we have that G = lim G =∼ Z . ←− W p Suppose now that p = 2; then A =∼ Z , for all W ∈ W.IfG acts freely on W 2 ∼ W DW ,forsomeW , then as in the case above, we get that GW = Z2. Since this also ∼ holds for every W ∈ W with W ≤ W , we deduce that G = Z2. So from now on we may assume that for every W ∈ W, there exists a vertex v0 = of DW such that (GW )v0 1. According to Lemma 4.2.6(d), every finite subgroup of GW has order at most 2; in particular, |(GW )v|≤2, for each v ∈ V(DW ).We deduce from Theorem 4.1.8 that G0 = (GW )v v ∈ V(DW ) is the closed subgroup of GW generated by its elements of order 2. By Corol- lary 4.1.3, GW /G0 is a free pro-2 group, since it is projective and pro-2. Therefore,

GW = G0 P, ∼ where P = GW /G0. By Lemma 4.2.6(e), = G0 CG0 (AW ) g , where g has order 2. By Lemma 4.2.6(c), CG0 (AW ) is a free pro-2 group. Since AW is normal in GW ,soisCG0 (AW ). Therefore, = CG0 (AW ), P CG0 (AW ) P is a torsion-free subgroup of GW . Since the stabilizers (GW )v (v ∈ V(DW ))are finite, it follows that the action of CG0 (AW ), P on DW is free. Then, by Theo- rem 4.1.2, CG0 (AW ), P is a projective group and so, being pro-2, it is a free pro-2 group. = = If CG0 (AW ) 1, we would have that G0 g ; this would imply, according to Proposition 4.2.2(b), that G0 acts trivially on DW , contradicting the fact that = =∼ GW , and so G0, acts faithfully on DW . Therefore CG0 (AW ) 1. Since AW Z2 ∩ is open in GW ,wehaveCG0 (AW ) AW is open and isomorphic to Z2; thus =∼ CG0 (AW ) Z2 (cf. RZ, Theorem 3.6.2(b)) and it is open in GW . Since P is free ∩ = = pro-p and CG0 (AW ) P 1, we deduce that P 1. So, = = =∼ GW G0 CG0 (AW ) g Z2 C2, where g acts on C (A ) =∼ Z by inversion, according to Lemma 4.2.6(d); i.e., ∼ G0 W 2 GW = Z2 C2 is the infinite dihedral pro-2 group. Next observe that if W1 ≤ W2 are in W, then the canonical map −→ GW1 GW2 is an epimorphism; but since Z2 C2 is Hopfian (see Sect. 1.3), this map is an isomorphism. Thus, G = lim G =∼ Z C . ←− W 2 2 4.2 Faithful and Irreducible Actions 129

Proposition 4.2.9 Let G be a pro-π group acting faithfully and irreducibly on a π-tree T . Assume that G contains a nontrivial abelian closed normal subgroup A. Then one of the following assertions holds: (a) G contains a nonabelian free pro-p subgroup acting freely on T , for some p ∈ π; (b) ∼ G = Z C = Zτˆ Zρˆ, ∼ ∼ where Z = Zτˆ , C = Zρˆ , τ and ρ are sets of prime numbers with τ ∩ ρ =∅; moreover G acts freely on T ; (c) ∼ G = Z C = Zτˆ Cn is a proﬁnite Frobenius group with Frobenius kernel Z and isolated Hall sub- ∼ ∼ group C, where Z = Zτˆ , C = Cn, τ is a set of prime numbers and n is a natural number; (d) ∼ G = Z C = Zτˆ C2, ∼ ∼ where Z = Zτˆ and C = C2, is an inﬁnite dihedral pro-τ group, i.e., 2 ∈ τ and C acts on Z by inversion.

Proof Using Zorn’s lemma, we may assume that A is a maximal abelian closed ∼ normal subgroup of G. By Lemma 4.2.6(a), A = Zσˆ , for some set of primes σ ⊆ π. Assume that case (a) does not hold, i.e., that G does not contain any nonabelian free pro-p subgroup acting freely on T , for any p ∈ π. We remark ﬁrst that if q is a prime number and a q-Sylow subgroup Gq of G is not procyclic, then Gq acts irreducibly on T . Indeed, if the action of Gq on T were reducible, then Gq ∩ A = 1, by Proposition 4.2.3(a); and so, Gq wouldbea(ﬁnite) cyclic group according to Lemma 4.2.6(b2), a contradiction.

Step 1. We claim that if q is a prime number and q>2, then any q-Sylow subgroup Gq of G is procyclic. If the action of Gq on T is irreducible, then by Proposi- ∼ tion 4.2.8 one has that Gq = Zq . Assume that Gq acts reducibly on T ; then Gq is a cyclic group, by the above remark. This proves the claim.

Step 2. Suppose that the 2-Sylow subgroups of G are also procyclic. Then G = K H , where K and H are procyclic groups of relatively prime orders (see Lemma 4.2.5). We may assume that K = 1. Applying Proposition 4.2.3(a) and (b), ∼ we deduce that K acts freely on T and K = Zτˆ , where τ ⊆ π.IfH = 1, then G = K has form (b) of the statement. So we may assume that H = 1. Now there are two possibilities: H acts either reducibly or irreducibly on T . In the ﬁrst case, one has ∼ that H = Cn, for some natural number n, by Lemma 4.2.6(b2), since K is abelian 130 4 Proﬁnite Groups Acting on C-Trees and normal in G; therefore G = K H has form (c) or (d) of the statement, accord- ingtoLemma4.2.6(d). In the second case, i.e., when H acts irreducibly on T ,it ∼ follows from Proposition 4.2.3(b) that H = Z ˆ , for some subset ρ of π. Therefore, ∼ ρ G = Zτˆ Zρˆ , with τ ∩ ρ =∅. We check next that in this case G acts freely on T . Indeed, let 1 = g ∈ G; then either g and K have coprime orders or K ∩ g=1. Using Lemma 4.2.6(b2) or Proposition 4.2.3(a), respectively, we deduce that g acts irreducibly on T ; thus g acts freely on T , by Proposition 4.2.3(b). Since g was chosen arbitrarily in G, we have that G acts freely on T . Hence G has form (b) of the statement.

Step 3. Assume now that the 2-Sylow subgroups of G are not procyclic; let L be a 2- Sylow subgroup of G. As we have remarked above, L acts irreducibly on T . Since G does not contain free nonabelian pro-p subgroups, it follows from Proposition 4.2.8 that L is isomorphic to an inﬁnite dihedral pro-2 group L = a b, ∼ ∼ where a = b = C2, the cyclic group of order 2, and denotes in this case the ∼ free pro-2 product of pro-2 groups. Put α = ab. Then L = α b = Z2 C2. Consider the centralizer CG(A) of A in G; since A is normal in G,sois CG(A). We claim that CG(A) = A. Indeed, observe ﬁrst that CG(A) is projective, by Lemma 4.2.6(c); so, by Step 1 and the form of the 2-Sylow subgroup of G, all the Sylow subgroups of CG(A) are procyclic. Hence CG(A) = D1 D2, where D1 and D2 are procyclic of relatively prime orders (see Lemma 4.2.5). Since D1 is a characteristic subgroup of CG(A), it is normal in G. Hence D1A is abelian and normal in G;soD1 ≤ A, by the maximality of A. It follows that D2 acts trivially on D1; therefore CG(A) is procyclic and, by the argument above,

CG(A) = A, since obviously CG(A) ≥ A. This proves the claim. Let ϕ : L = a b−→Aut(A) be the homomorphism given by conjugation. One easily deduces from Lemma 4.2.6(d) that conjugation of elements in A by a or b is inversion, i.e., axa = bxb = x−1, for all x ∈ A: indeed, just replace A by any of its p-Sylow subgroups Ap, and observe that Aut(Ap) has a unique subgroup of order 2 (cf. RZ, Theorem 4.4.7). Therefore the image Im(ϕ) of ϕ is a group of order 2. Hence

Ker(ϕ) = CL(A) = CG(A) ∩ L = A ∩ L has index 2 in L. Since L has a unique torsion-free normal subgroup of index 2, we deduce that ∼ A ∩ L = α = Z2, ∼ and thus Z2 = α=A2, the (unique) 2-Sylow subgroup of A = CG(A). 4.2 Faithful and Irreducible Actions 131

Consider next the continuous homomorphism

ψ : G −→ Aut(A2) = Z2 × C2 ∼ given by conjugation. By an argument similar to the one used above, ψ(L) = C2. Since ψ(G) is a pro-2 group and L is a 2-Sylow subgroup of G,wehavethat ∼ ψ(G)= ψ(L)= C2. Thus Ker(ψ) = CG(A2) has index 2 in G. Choose an element g ∈ G of order 2. Since CG(A2) is projective, and so torsion-free, g/∈ CG(A2); therefore,

G = CG(A2) g.

Now, since CG(A2) ∩ L is normal, torsion-free and of index 2 in L,wehave CG(A2) ∩ L = α=A2. We deduce that A2 is also the 2-Sylow subgroup of CG(A2). Therefore all Sylow subgroups of CG(A2) are cyclic, and so CG(A2) is a semidirect product of two procyclic groups of relatively prime orders (see Lemma 4.2.5). It follows that

CG(A2) = A2 × H, for some subgroup H .IfH = 1, then G = A2 g, a dihedral pro-2 group. Assume H = 1. We claim that H is procyclic. Since the order of H is not divisible by 2, H is characteristic in CG(A2) and so it is normal in G. Hence H acts irreducibly on T , according to Proposition 4.2.3. To prove the claim we study the subgroup H g=H g. Since the Sylow subgroups of H g are procyclic, we have that H g=M N, where M and N are procyclic of relatively prime orders (see Lemma 4.2.5). Clearly, g/∈ M, for otherwise g would be normal in M, and so it would act irreducibly (and faithfully) on T , by Proposition 4.2.3(a), contradicting part (b) of that proposition. Replacing N by one of its conjugates in H g, we may assume that g ∈ N.We prove next that in fact N = g. Indeed, if this were not the case, then N would contain a subgroup C isomorphic to Zp, for some prime p = 2; this would imply ∼ that C acts irreducibly on T , by Lemma 4.2.6(b2), since A2 = Z2 is normal in G; finally, since C g acts irreducibly and faithfully on T (see Remark 4.2.1(b)), we can apply Lemma 4.2.6(d) (1) (with C g playing the role of G and C playing the role of A in that lemma), to deduce that C g is a Frobenius group with isolated subgroup g; this contradicts the fact that C g is contained in N, which is procyclic. Thus N = g. Therefore, H g=M g and both H and M are normal Hall subgroups of the prosolvable subgroup H g; thus H = M (cf. RZ, Corollary 2.3.7), proving the claim. ∼ Thus CG(A2) = A2 × H = Z = Zτˆ is procyclic, and G = Z g is an infinite dihedral profinite group, since 2 ∈ τ (see Lemma 4.2.6(d)).

We prove next the promised structure theorem for profinite groups that act faithfully and irreducibly on a profinite tree. The theorem has a Tits alternative flavor: it says that a pro-π group G acting faithfully and irreducibly on a π-tree T either contains a free nonabelian pro-p subgroup or it is solvable of a very special type. 132 4 Profinite Groups Acting on C-Trees

Theorem 4.2.10 Let G be a nontrivial pro-π group acting faithfully and irreducibly on a π-tree T . Then one of the following assertions holds: (a) G contains a nonabelian free pro-p subgroup acting freely on T , for some p ∈ π. (b) G is solvable and has one of the following forms:

(b1) ∼ G = Z C = Zσˆ Zρˆ, ∼ ∼ where Z = Zσˆ , C = Zρˆ , and σ and ρ are disjoint sets of prime numbers; moreover G acts freely on T ; (b2) ∼ G = Z C = Zσˆ Cn ∼ is a proﬁnite Frobenius group with Frobenius kernel Z, where Z = Z ˆ , ∼ σ C = Cn, σ is a set of prime numbers and n is a natural number; (b3) ∼ G = Z C = Zσˆ C2, ∼ ∼ where Z = Zσˆ and C = C2, is an inﬁnite dihedral pro-σ group, i.e., 2 ∈ σ and C acts on Z by inversion.

Proof Decompose G as an inverse limit as in Lemma 4.2.7. We continue with the notation in that lemma. Since G = 1, we have that GW = 1, for every W ∈ W. Suppose that (a) does not hold. It follows from Lemma 4.2.7(c) that GW contains a closed normal subgroup isomorphic to Zσˆ , where ∅=σ ⊆ π, and GW does not contain free nonabelian pro-p subgroups, for each W ∈ W. By Proposition 4.2.9, each GW is isomorphic to a group of the form (b1), (b2) or (b3). Note that a group of type (b1) cannot be a quotient group of a group of type (b3), because a group of type (b1) is torsion-free and so it is not generated by elements of order 2. On the other hand, a 2-Sylow subgroup of a group of type (b2) has order 2, while a 2-Sylow subgroup of a group of type (b3) has order 2∞; therefore, a group of type (b3) cannot be a quotient group of a group of type (b2). Thus replacing the inverse system {GW | W ∈ W} by a coﬁnal subset if necessary, we may assume that the GW are all of type (b1) or type (b2) or type (b3). If all the GW (W ∈ W) are of type (b1), then = G lim←− GW W∈W is torsion-free and its p-Sylow subgroups are procyclic, for every prime p.SoG is of type (b1) according to Lemma 4.2.5. Assume that all the GW (W ∈ W) are of type (b2). Say

GW = ZW CW 4.2 Faithful and Irreducible Actions 133 is a Frobenius group with isolated (ﬁnite cyclic) group CW , and with Frobenius kernel ZW .LetW,W ∈ W with W ≤ W . Consider the canonical epimorphism ϕW,W : GW −→ GW . Since ϕW,W (ZW ) has ﬁnite index in GW ,wehavethat ZW ∩ ϕW,W (ZW ) = ∅. Since the centralizer of a nontrivial element of ZW ∩ ϕW,W (ZW ) is in ZW (cf. RZ, Theorem 4.6.9(e)), we deduce that ϕW,W (ZW ) ≤ ZW .Let = Z lim←− ZW . W∈W

Then Z is a closed normal Hall subgroup of G, and CG(k) ≤ Z, for every 1 = k ∈ Z, since the same properties hold for each ZW in GW (W ∈ W), we have that G is a proﬁnite Frobenius group with Frobenius kernel Z (cf. RZ, Theorem 4.6.9). By ∼ Proposition 4.2.3(b), Z = Zσˆ , for some nonempty set of primes σ . Finally, assume that each GW is of type (b3), i.e., it is an inﬁnite dihedral pro-σW group, say, GW = ZW gW , where σW is a set of prime numbers containing 2, =∼ ZW Zσ W and gW is an element of order 2, for each W . Note that

GW = ZW ∪. ZW gW , and ZW gW is the set of elements of order 2 in GW . Consider the canonical epimorphism ϕW,W : GW −→ GW , where W,W ∈ W with W ≤ W . Observe that ϕW,W (ZW gW ) ⊆ ZW gW , because if ϕW,W (gW ) = 1, then GW would be procyclic. Since an element of order 2 of GW acts on ZW by inversion, we deduce that ϕW,W (ZW ) ≤ ZW .Let = ; Z lim←− ZW W∈W then Z is a procyclic group whose order is divisible by 2 and Z is normal in G; ∼ furthermore, according to Proposition 4.2.3(b), Z = Zσˆ , for some set of primes σ . Since ZW gW is compact for each W ∈ W, there exists an element ∈ g lim←− ZW gW W∈W (see Sect. 1.1), and g has order 2. It follows that G = Z g=Z g is a dihedral pro-σ group.

We can now prove the main theorem of this section.

Theorem 4.2.11 Let G beapro-π group acting on a π-tree T . Then one of the following assertions holds:

(a) G = Gv is the stabilizer of a vertex v of T . (b) G has a free nonabelian pro-p subgroup P such that P ∩ Gv = 1, for every vertex v of T . (c) There exists an edge e of T whose stabilizer Ge is normal in G, and the quotient group G0 = G/Ge is solvable of one of the following types: 134 4 Proﬁnite Groups Acting on C-Trees

(c1) ∼ G0 = Z C = Zσˆ Zρˆ, ∼ ∼ where Z = Zσˆ , C = Zρˆ , and σ and ρ are disjoint sets of prime numbers; in particular G0 is a projective group; (c2) ∼ G0 = Z C = Zσˆ Cn is a proﬁnite Frobenius group with Frobenius kernel Z and isolated sub- ∼ ∼ group C, where Z = Zσˆ , C = Cn, σ is a set of prime numbers and n is a natural number; (c3) ∼ G0 = Z C = Zσˆ C2, ∼ ∼ is an inﬁnite dihedral pro-σ group, where Z = Zσˆ and C = C2,2∈ σ and C acts on Z by inversion.

Proof Choose a minimal G-invariant π-subtree D of T (see Proposition 2.4.12). Let

CG(D) ={x ∈ G | xm = m, ∀m ∈ D} be the kernel of the action of G on D, and set G0 = G/CG(D). Then G0 acts faithfully and irreducibly on D. By Theorem 4.2.10, one of the following assertions holds for G0:

(α) G0 ={1}; (β) G0 contains a nonabelian free pro-p subgroup acting freely on D,forsome prime number p; ∼ ∼ ∼ (γ ) G0 = Z C = Zσˆ Zρˆ , where Z = Zσˆ , C = Zρˆ , σ and ρ are sets of prime numbers and σ ∩ ρ =∅; ∼ (δ) G0 = Z C = Z ˆ Cn is a proﬁnite Frobenius group with Frobenius kernel Z σ ∼ ∼ and isolated subgroup C, where Z = Zσˆ , C = Cn, σ is a set of prime numbers and n is a natural number; ∼ ∼ ∼ (ε) G0 = Z C = Zσˆ C2, where Z = Zσˆ and C = C2, is an inﬁnite dihedral pro-σ group, i.e., 2 ∈ σ and C acts on Z by inversion.

We consider each of these cases in turn. If G0 is trivial, then |D|=1. Say D ={v}. Then G = Gv; that is, case (a) of the theorem holds. Consider now case (β). Let H be a nonabelian free pro-p subgroup of G0 acting freely on D. Since H is projective, G has a subgroup P which is mapped isomorphically onto H by the natural homomorphism G −→ G0. Therefore, P acts freely on D, that is, P ∩ Gv ={1}, for all v ∈ V(D). So, by Corollary 4.1.9, P ∩ Gv ={1}, for all v ∈ V(T), and hence P is a free pro-p group, i.e., case (b) of the theorem holds. 4.2 Faithful and Irreducible Actions 135

Assume next that G0 = 1 and that G0 does not contain nonabelian free pro-p subgroups acting freely on D, i.e., neither (α) nor (β) hold; in particular |D| > 1, hence D has at least one edge (see Proposition 2.1.6(c)). We claim that then

CG(D) = Ge, for some edge e of D. Observe that if this claim holds, the proof of the theorem will be complete, since cases (γ ), (δ),(ε) above correspond to items (c1), (c2), (c3) of the theorem. To verify the claim it is enough to show that there exists an edge e ∈ E(D) with trivial G0-stabilizer. In case (γ ), G0 = G/CG(D) acts freely on D, and so the claim holds in this case. In the remaining of the proof we assume that we are in either case (δ)or(ε). Set ˜ D = d ∈ D (G0)d = 1 . ˜ Observe that D = ∅, because G0 = 1 and G0 acts faithfully on D. Next we assert that D˜ is closed in D. In these two cases the normal subgroup Z of G0 acts freely on D (see Proposition 4.2.3(b)). Therefore, since Z is open in G0, we deduce that for each element d of D, the stabilizer (G0)d of d in G0 is ﬁnite. L For a subgroup L of G0, denote by D the set of ﬁxed points of D under the action of L. Put L DC = D . {1}=L≤C

In case (δ), every finite subgroup of G0 is contained in some conjugate of C (cf. RZ, Corollary 2.3.7). So, d ∈ D˜ if and only if there exists a nontrivial subgroup L zLz−1 L ˜ of C and some z ∈ Z such that d ∈ D = zD . Hence D = ZDC . Thus, since each DL is closed and since C is finite, we have that D˜ is a closed subset of D, proving the assertion in case (δ). In case (ε),letz and c be generators of Z and C, respectively. One checks easily that there are exactly two conjugacy classes in G0 of subgroups of finite order, namely the class of C = c and the class of zc. So, in this case, D˜ = ZD c ∪ ZD zc. Thus, we have that D˜ is a closed subset of D, proving the assertion in case (ε).So in either case, D˜ is a profinite graph. Next we prove that D˜ is disconnected. In case (ε), we have that ZD c = ∅ = ZD zc by Theorem 4.1.8. Note that ZD c ∩ ZD zc =∅: indeed, otherwise there c zc exist k ∈ Z, d ∈ D and d ∈ D such that kd = d ; hence zc=(G0)d = −1 −1 ˜ k(G0)d k = k ck , a contradiction. This implies that D is not connected (see Lemma 2.1.9). In case (δ), choose a proper open subgroup Z of Z. Then one has ˜ D = Z DC ∪ (Z − Z )DC . Assume that Z DC ∩ (Z − Z )DC = ∅; then there exist d,d ∈ DC and k ∈ Z − Z (in particular k = 1), such that d = kd and Cd = −1 −1 1 = Cd . Since Cd = Ckd = kCd k , we deduce that C ∩ kCk = 1, contradicting the fact that the group C is isolated in G0. Therefore Z1DC ∩ (Z − Z1)DC =∅. Thus, using again Lemma 2.1.9, we obtain that D˜ is not connected. 136 4 Profinite Groups Acting on C-Trees

Now, since D˜ is not connected, there exists an edge e ∈ D − D˜ , because otherwise E(D)˜ = E(D) and so D˜ would coincide with D in every ﬁnite quotient of D, since D is connected; but D˜ is closed, so this would imply that D˜ = D, a contradiction. Then (G0)e ={1}. This veriﬁes the claim, and the theorem is proved. Chapter 5 Free Products of Pro-C Groups

5.1 Free Pro-C Products: The External Viewpoint

Throughout this section C denotes a pseudovariety of ﬁnite groups unless otherwise indicated.

Let T be a profinite space. A sheaf of pro-C groups over T is a triple (G,π,T), where G is a profinite space and π : G −→ T is a continuous surjection satisfying the following conditions: (a) for every t ∈ T ,thefiber G(t) = π −1(t) over t is a pro-C group whose topology is induced by the topology of G; (b) multiplication and inversion in each fiber G(t) depends continuously on t.More precisely, if we define G2 = (g, h) ∈ G × G π(g) = π(h) ,

then the map 2 μG : G −→ G, − given by μG(g, h) = gh 1, is continuous.

G G(t)

π T • t

Observe that the space G2 is closed as a subspace of G × G, and therefore it is a proﬁnite space. We often write G rather than (G,π,T), if the context is clear.

© Springer International Publishing AG 2017 137 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_5 138 5 Free Products of Pro-C Groups

Example 5.1.1 (See also Sect. 5.7)

(a) Let T ={1,...,n} be a finite discrete space with n points and let G1,...,Gn be pro-C groups. Define the space G = G1 ∪. ···∪. Gn to have the disjoint topology. Let π : G → T be the map that sends Gi to i (i = 1,...,n). Then (G,π,T) is in a natural way a sheaf over the space T with G(i) = Gi (i = 1,...,n). (b) Let G be a profinite group and let T be a profinite space. Define the constant sheaf over T with constant fiber G to be the sheaf

KT (G) = (T × G, π, T ),

where π : T × G −→ T is the usual projection map. (c) Let S be a set (not necessarily finite), which we consider as a discrete space. Let T = S ∪. {∗} be its one-point compactification. Assume that {Gt | t ∈ T } is a collection of pro-C groups such that G∗ = 1. Define a topology on G = . ∈ = ∗ t∈T Gt as follows: if x Gt , t , then the set of all neighbourhoods of x in Gt is a fundamental system of neighbourhoods of x in G; and a fundamental system of neighbourhoods of x =∗in G is given by the subsets of the form ∗ t∈U Gt , where U ranges through the basic open neighbourhoods of in T , i.e., each U is a subset of T containing ∗ such that T − U is finite. Define π : G −→ T by π(Gt ) = t. Then (G,π,T)is a sheaf, which we term a sheaf of groups over S converging to 1.

A morphism

α = α, α : (G,π,T)−→ G ,π ,T of sheaves of pro-C groups consists of a pair continuous maps α : G −→ G and α : T −→ T such that the diagram

α G G

π π α T T commutes and the restriction of α to G(t) is a homomorphism from G(t) into G (α (t)), for each t ∈ T . We say that α = (α, α ) is a monomorphism (respectively, epimorphism)ifα and α are injective (respectively, surjective). The image of a monomorphism (respectively, epimorphism) G −→ G is called a subsheaf (respectively, a quotient sheaf ) of the sheaf G (respectively, G). Note that if (G,π,T) is a sheaf and T is a closed subspace of T , then the triple

−1 π T ,π|π−1(T ),T is a subsheaf of (G,π,T). 5.1 Free Pro-C Products: The External Viewpoint 139

Let (G,π,T) be a sheaf of pro-C groups and let H beapro-C group. We may think of H as the fiber of a sheaf over a singleton space. Hence we have a natural notion of a morphism α : G −→ H from the sheaf G to the group H , namely, α is a continuous map from the space G into H such that the restriction of α to each fiber G(t) is a homomorphism. Let (G,π,T) be a sheaf of pro-C groups. A free pro-C product of the sheaf G is defined to be a pro-C group G together with a morphism ω : G −→ G,having the following universal property: for every morphism β of the sheaf G into a pro-C group H , there exists a unique continuous homomorphism β¯ : G −→ H such that the following diagram

ω G G(t) G

β¯ π β T • H t is commutative: βω¯ = β. Since a pro-C group H is an inverse limit of groups in C, it is sufﬁcient to check the above universal property for groups H ∈ C only.

Proposition 5.1.2 Let (G,π,T)be a sheaf of pro-C groups. Then there exists a free pro-C product of G and it is unique up to isomorphism.

Proof The uniqueness follows easily from the universal property. We shall give an explicit construction of the free pro-C product. Let

L = ∗ G(t) t∈T be the free product of the groups G(t), t ∈ T , considered as abstract groups. Let ρ : G −→ L be deﬁned on each G(t) as the inclusion map. Consider the set N of all normal subgroups N of L with L/N ∈ C such that the composite map

ρ G −→ L −→ L/N is continuous. One checks that if N1,N2 ∈ N , then N1 ∩N2 ∈ N . So, L can be made into a topological group by considering N as a fundamental system of neighbourhoods of the identity element of L. Denote by KN (L) the corresponding completion of L with respect to this topology K = N (L) lim←− L/N. N∈N 140 5 Free Products of Pro-C Groups

Then KN (L) is a pro-C group. Let

ι : L −→ KN (L) be the natural map. Put ω = ιρ ; then ω is continuous because the composite

G −→ KN (L) −→ L/N is continuous for each N ∈ N . Since the restriction of ω to each G(t) (t ∈ T)is a homomorphism, the map ω : G → KN (L) is a morphism. We claim that (KN (L), ω) is a free pro-C product of G. To see this we check the corresponding universal property. Let H ∈ C and let β : G −→ H be a morphism.

KN (L) β¯ ι β˜ ω L H

ρ β G

By the universal property of abstract free products, there exists a unique homomorphism β˜ : L −→ H with β = βρ˜ . Since βρ˜ is continuous, it follows from the deﬁnition of N that Ker(β)˜ ∈ N ; so there exists a continuous homomorphism β¯ : KN (L) −→ H with β˜ = βι¯ . Hence β = βω¯ . The uniqueness of β¯ follows from the fact that KN (L) = ω(G).

The free pro-C product of a sheaf (G,π,T)will be denoted by

C G or simply by G T T if there is no danger of confusion and the pseudovariety C is understood.

Example 5.1.3

(a) Let T ={1,...,n} be a ﬁnite discrete space with n points and let G1,...,Gn be pro-C groups. Consider the sheaf (G,π,T)deﬁned in Example 5.1.1(a): G is C just the disjoint! union of the groups G1,...Gn. The corresponding free pro- product G = G coincides with the standard concept of a free pro-C product of the groups Gi (see Sect. 1.6), and it is usually written n G = G1 ···Gn = Gi i=1 5.1 Free Pro-C Products: The External Viewpoint 141

C C (or G = G1 ··· Gn, if one needs to emphasize the pseudovariety C). Indeed, if ω : G → G is the canonical morphism, define ωi : Gi → G to be the restriction of ω!to Gi (i = 1,...,n); then the corresponding universal property defining G = G is the following: for any given pro-C group H and continuous homomorphisms ϕi : Gi → H (i = 1,...,n), there exists a unique continuous homomorphism ϕ : G → H such that ϕωi = ϕi (i = 1,...,n). (b) Let T be a profinite space. Consider the constant sheaf

KT (ZCˆ) = (T × ZCˆ,π,T) ! (see Example 5.1.1(b)). Then the free pro-C product KT (ZCˆ) is isomorphic to the free pro-C group FC(T ) on the profinite space T . (c) Let (T , ∗) be a pointed profinite space and let (H,π,T)be a sheaf whose fibers are Z ˆ, if t ∈ T −{∗}; H(t) = C ∗, if t =∗, . so that H =[(T −{∗}) × ZCˆ]∪{∗}. Define the map π : H −→ T by setting π(t,z) = t,if∗=t ∈ T , and π(∗) =∗. The topology on H is just the product −{∗} × ∗ H topology on (T ) ZCˆ and basic open neighbourhoods of in are the H ∪. {∗} subsets of the form t∈U−{∗} (t) , where U is an open neighbourhood of ∗ in T!. Then one checks that (H,π,T) is indeed a sheaf and the free pro-C H C ∗ product T is isomorphic to the free pro- group FC(T , ) on the pointed space (T , ∗).

Next we record a series of properties of external free products.

Lemma 5.1.4 Let (G,π,T) be a sheaf of pro-C groups and let ν : G −→ K be a continuous map into a group K ∈ C such that its restriction νs : G(s) −→ K to a ﬁber G(s) (for some ﬁxed s ∈ T ) is a homomorphism. Then there exists a neighbourhood S of s in T such that the restriction νt : G(t) −→ K of ν to G(t) is a homomorphism for every t ∈ S.

Proof Deﬁne maps

2 η : G −→ K × K, (g1,g2) → ν(g1)ν(g2), ν(g1g2) , and 2 ρ : G −→ T, (g1,g2) → π(g1) = π(g2). It is easy to see that these maps are continuous. Put = (a, b) ∈ K × K a = b .

It follows that η−1() is open (and closed). Next we claim that S = t ∈ T ρ−1(t) ⊆ η−1() 142 5 Free Products of Pro-C Groups is open in T . To see this it sufﬁces to prove that T − S is closed, and this follows from two facts: on the one hand G2 −η−1() is compact since G2 is closed in G ×G, and on the other hand

T − S = ρ G2 − η−1() .

Finally, observe that t ∈ S if and only if ν is a homomorphism on G(t).

Corollary 5.1.5 Let (G,π,T) be a sheaf of pro-C groups and let s,t ∈ T with s = t. Then every continuous homomorphism σs : G(s) −→ K into a group K ∈ C can be extended to a morphism σ : G −→ K such that σ(G(t)) = 1.

∈ −1 G Proof For each k K, σs (k) is compact. Since is a proﬁnite space, there ex- G ∈ −1 ⊆ ∈ ist clopen subsets Uk of (k K) such that σs (k) Uk, for all k K, and

Uk ∩ Uk =∅, whenever k,k ∈ K and k = k . Deﬁne ν : G −→ K by k, if x ∈ U ; ν(x) = k ∈ 1, if x/ k∈K Uk.

Then ν is continuous and it coincides with σs on G(s). Now it follows from Lemma 5.1.4 that there is a clopen neighbourhood V of s in T such that t/∈ V and ν is a homomorphism when restricted to G(r), for every r ∈ V . Finally, de- ﬁne σ : G −→ K to coincide with ν on π −1(V ), and such that σ(x) = 1if x ∈ G − π −1(V ). Since π −1(V ) is clopen, σ is continuous. Clearly σ is a homomorphism when restricted to each ﬁber G(r), and it extends σs . ! G C = C G Proposition 5.1.6 Let ( ,π,T) be a sheaf of pro- groups, G T , and let ω : G −→ G be the canonical morphism. Then:

(a) the group G is generated by its subgroups Gt = ω(G(t)), t ∈ T ; (b) if s = t, then Gs ∩ Gt = 1; (c) the morphism ω maps G(t) isomorphically onto Gt , for all t ∈ T . ! = C G Proof The explicit construction of G T in the proof of Proposition 5.1.2 shows that (a) holds. To prove (b) and (c) we may assume that |T | > 1. Let s,t ∈ T with s = t, and let 1 = g ∈ G(t).LetU be an open normal subgroup of G(t) such that g/∈ U. Put K = G(t)/U. Then, by Corollary 5.1.5, there exists a morphism σ : G −→ K extending the natural epimorphism G(t) −→ K and such that σ(G(s)) = 1. By the universal property of a free product, there exists a continuous homomorphism σ¯ : G −→ K such that σω¯ = σ . Since σ(g) = 1, we deduce that ω(g) = 1. Therefore ω is a monomorphism on G(t); this proves part (c). Since σ(G(s)) = 1, it follows that ω(g)∈ / Gs . Since g is a nontrivial arbitrary element of G(t), this means that Gs ∩ Gt = 1, proving part (b). 5.1 Free Pro-C Products: The External Viewpoint 143

Let {(G ,π ,T ), ϕ ,I} be an inverse system of sheaves of pro-C groups over a i i i ij directed poset (I, ), where ϕ = (ϕ ,ϕ ), ij ij ij : G −→ G : −→ ϕij i j and ϕij Ti Tj (i j). Let G = G lim←− i i∈I be its inverse limit. Observe that this is a sheaf over the space T = lim ∈ T , and ←− i I i whose ﬁber over t = (t ) ∈ T is G(t) = lim ∈ G (t ): indeed, the multiplication map i ←− i I i i

μG : G2 −→ G is continuous, because μG = lim ∈ μG . Denote by ϕ = (ϕ ,ϕ ) : ←− i I i i i i G −→ Gi the projection (i ∈ I). The next proposition shows that the operations of taking free products and inverse limits commute.

Proposition 5.1.7 With the above notation, G = G lim←− i lim←− i. ∈ ∈ T i I i I Ti

Before we prove this proposition we need the following lemma, where we continue with the above notation.

Lemma 5.1.8 Every morphism : G = G −→ β lim←− i H i∈I into a discrete ﬁnite group H ∈ C can be factored through a projection

ϕ : G −→ G , k k for some k ∈ I , i.e., there exists some k ∈ I and some morphism β : Gk −→ H such that β = β ϕ . k

Proof By assumption β is a continuous map from the topological space G into the discrete finite group H such that the restriction of β to each fiber G(t) is a homomorphism (t ∈ T ). Since G = lim ∈ G as topological spaces, β factors through a contin- ←− i I i : G −→ ∈ uous function βi0 i0 H ,forsomei0 I (see Sect. 1.3). However, βi0 need not G G ={ ∈ | } be a homomorphism when restricted to a fiber i0 (t) of i0 . Put I0 i I i i0 . For each i ∈ I , define β : G −→ H by β = β ϕ ; then clearly β = β ϕ .We 0 i i i i0 ii0 i i claim that for some k ∈ I0,themapβk is a morphism, i.e., its restriction to each fiber Gk(t) is a homomorphism. To see this consider the continuous map

2 η : G −→ H × H, (g1,g2) → β(g1)β(g2), β(g1g2) , 144 5 Free Products of Pro-C Groups

: G2 −→ × ∈ and the analogous continuous maps ηi i H H , for each i I0. It is easy to check that G2 = G2 = ; lim←− i and η lim←− ηi i i0 i i0 furthermore, G2 = G2 = G2 η lim←− ηi i ηi i .

i i0 i 0 G2 × Since ηi( i ) is contained in the finite set H H and I0 is a directed poset, it follows G2 = G2 ∈ that η( ) ηk( k ),forsomek I0. Next observe that since β is a morphism, 2 2 η(G ) ⊆ ={(h, h) | h ∈ H}. Therefore ηk(G ) ⊆ . Thus ηk is a homomorphism k when restricted to each fiber Gk(t). Put β = ηk. ! Proof of Proposition 5.1.7 Let Gi = Gi , with universal morphism ωi : Gi −→ Gi (i ∈ I ). If i, j ∈ I , i j, define ψij : Gi −→ Gj to be the continuous homomorphism induced by the morphism ωj ϕij : Gi −→ Gj , using the universal property of free products. Then {Gi,ψij ,I} is an inverse system over I . Define

= = : G −→ G lim←− Gi and ω lim←− ωi G, i∈I i∈I ! and let ψi : G −→ Gi denote the projection (i ∈ I). We shall prove that G = G with universal morphism ω. Let H ∈ C and let β : G −→ H be a morphism. By Lemma 5.1.8, there exists ¯ some k ∈ I and a morphism βk : Gk −→ H such that β = βkϕk.Letβk : Gk −→ H ¯ ¯ be the continuous homomorphism induced by βk so that βk = βkωk. Deﬁne β = ¯ ¯ ¯ βkϕk. Then clearly β = βω. The uniqueness of the continuous homomorphism β satisfying this last equality follows from the fact that ω(G) generates G, because ωi(Gi) generates Gi (i ∈ I), according to part (a) of Proposition 5.1.6.

Example 5.1.9 Let S be a set and let {Gs | s ∈ S} be a collection of pro-C groups. Consider the corresponding sheaf (G,π,T) of groups over S converging to 1 as in Example 5.1.1(c). Let G =G be its corresponding free pro-C product. If H is a pro-C group, and ϕs : Gs → H is a continuous homomorphism for each s ∈ S,we say that the collection of homomorphisms {ϕs | s ∈ S} converges to 1 if whenever U is an open neighbourhood of 1 in H , then U contains ‘almost all’ (= all but a = ﬁnite number of) the subgroups ϕs(Gs) of H . Observe that if we deﬁne ωs ω|Gs , then the collection {ωs | s ∈ S} converges to 1. It is easy to check that G =G is characterized by the following universal property: whenever {ϕs : Gs → H | s ∈ S} is a collection of continuous homomorphisms into a pro-C group H converging to 1, then there exists a unique continuous homomorphism ϕ : G →!H such that = ∈ C r ϕωs ϕs , for all s S. This type of free pro- product is denoted by s∈S Gs , and it is called the restricted free pro-C product of the collection {Gs | s ∈ S}. 5.2 Subgroups Continuously Indexed by a Space 145

5.2 Subgroups Continuously Indexed by a Space

Let T be a proﬁnite space and let G be a proﬁnite group. A collection of closed subgroups Gt of G(t∈ T)is said to be continuously indexed by T if whenever U is an open subset of G, then the subset

T(U)={t ∈ T | Gt ⊆ U} of T is open in T . For a proﬁnite group G, deﬁne Subgp(G) to be the set of all closed subgroups of G. The collection Subgp(U) U ≤o G of subsets of Subgp(G), for all open subgroups U of G, is a base for a topology on Subgp(G). This is because if U1,...,Un are open subgroups of G, then

Subgp(U1 ∩···∩Un) = Subgp(U1) ∩···∩Subgp(Un).

This topology is called the étale topology of Subgp(G).

Lemma 5.2.1 Let G be a proﬁnite group and let {Gt | t ∈ T } be a collection of closed subgroups of G indexed by a proﬁnite space T . The following conditions are equivalent:

(a) The family {Gt | t ∈ T } is continuously indexed by T . (b) The set E = (t, g) ∈ T × G t ∈ T,g∈ Gt is a closed subset of T × G. (c) The function ϕ : T −→ Subgp(G),

given by ϕ(t) = Gt , is continuous, where Subgp(G) is endowed with the étale topology . = (d) D t∈T Gt is a closed subset of G.

Proof Assume ﬁrst that the family {Gt | t ∈ T } is continuously indexed by T .Let (t, g) ∈ (T × G) − E; then g/∈ Gt . Hence, by the compactness of Gt , there exist open subsets U and V of G such that g ∈ U, Gt ⊆ V and U ∩ V =∅. Since the subgroups Gs are continuously indexed by T , T(V)={s ∈ T | Gs ⊆ V } is open in T .SoT(V)× U is an open neighbourhood of (t, g) in T × G, and (T (V ) × U)∩ E =∅. Hence E is closed in T × G. Conversely, assume that E is closed. Let U be an open neighbourhood of 1 in G. Let π : T × G −→ T denote the projection. Then

T − T(U)= π E ∩ T × (G − U) . 146 5 Free Products of Pro-C Groups

Therefore, T − T(U)is compact. Hence T(U)is open. So the family

{Gt | t ∈ T } is continuously indexed by T . This proves the equivalence of (a) and (b). Obviously (a) implies (c). Assume now that (c) holds, that is, assume that ϕ is continuous. Let A be an open subset of G. Deﬁne Sgp(A) = H ∈ Subgp(G) H ⊆ A .

Note that Sgp(A) = Subgp(H U) H ∈ Subgp(G), U o G, H U ⊆ A ; so Sgp(A) is open in Subgp(G). Since T(A)= ϕ−1(Sgp(A)), T(A)is open in T . Hence (a) holds. If (b) holds, then E is compact. Therefore so is D = γ(E), where

γ : T × G −→ G is the projection map. Hence (b) implies (d). Conversely, assume that (d) holds. Let (t, g) ∈ T × G − E (t ∈ T,g ∈ G). Then g/∈ D. Therefore, there exists an open neighbourhood V of g in G such that V ∩ D =∅. It follows that T × V is an open neighbourhood of (t, g) in T × G missing E. Thus (b) holds.

A standard way of obtaining a collection of continuously indexed closed subgroups of a proﬁnite group is recorded in the following lemma.

Lemma 5.2.2 Let G be a proﬁnite group that acts continuously on a proﬁnite space T . Then the collection {Gt | t ∈ T } of the G-stabilizers Gt of the points t of T is continuously indexed by T .

Proof Consider the map α : T × G −→ T × T deﬁned by α(t,g) = (t, gt) (g ∈ G, t ∈ T). Since α is continuous and the diagonal subset ={(t, t) | t ∈ T } of T × T is closed, it follows that −1 E = (t, g) ∈ T × G g ∈ Gt = α () is closed in T × G. The result now follows from Lemma 5.2.1.

Proposition 5.2.3 Let G be a proﬁnite group and let

F ={Gr | r ∈ T } 5.2 Subgroups Continuously Indexed by a Space 147 be a continuously indexed family of closed subgroups of G, where T is a proﬁnite space. Consider the equivalence relation on the product space T × G deﬁned by

−1 (t, g) ∼ t ,g if t = t and g g ∈ Gt t,t ∈ T ; g,g ∈ G .

Then the quotient space G/F = T × G/∼ is a proﬁnite space.

Proof Let ϕ : T × G −→ G/F be the canonical quotient map. Observe that, as sets, . G/F = (G/Gr ). r∈T

With this identiﬁcation, ϕ(t,g) = gGt ∈ G/Gt (t ∈ T,g∈ G). For

(t, g) ∈ T × G, we also write (t, g) = ϕ(t,g). Clearly G/F is compact. We shall show that this space is also Hausdorff and totally disconnected. Let (t1,g1) = (t2,g2) in G/F. We shall prove that there exist disjoint open neighbourhoods in G/F of these two points. Note that if g ∈ G and t ∈ T , then

−1 ϕ (t, g) ={t}×gGt ⊆ T × G.

Case 1. Assume that t1 = t2. Choose clopen neighbourhoods U1 and U2 of t1 and t2, −1 respectively, in T , with U1 ∩ U2 =∅. Then Ui × G = ϕ (ϕ(Ui × G)), and hence ˜ Ui = ϕ(Ui × G) = G/Gr r∈Ui ˜ ˜ ˜ is open in G/F (i = 1, 2). Note that (ti,gi) ∈ Ui (i = 1, 2) and U1 ∩ U2 =∅. = = = −1 ∈ Case 2. Assume that t1 t2. Put t t1 t2. Then g1 g2 / Gt . So, since Gt and G are proﬁnite groups, there exists an open normal subgroup W of G such that −1 ∩ =∅ { | ∈ } Wg1 g2 WGt . Since the family Gr r T is continuously indexed, there −1 ∩ =∅ exists a clopen neighbourhood M of t in T such that Wg1 g2 Gs , for all s ∈ M. By continuity of multiplication in G, there exists an open subgroup U of G −1 such that (g1u1) (g2u2)/∈ Gs , for all u1,u2 ∈ U and s ∈ M. Put ˜ Ui = ϕ(M × giU)= gGs,(i= 1, 2). g∈gi U,s∈M ˜ ˜ −1 ˜ ˜ It follows that U1 ∩ U2 =∅.Now,M × giU = ϕ (Ui), and so Ui is open in G/F ˜ ˜ (i = 1, 2). Hence U1 and U2 are open neighbourhoods of (t1,g1) and (t2,g2),respectively, in G/F, and they are disjoint. This proves that G/F is a Hausdorff space. It then follows from our deﬁnition ˜ ˜ of U1, in either case, that U1 is a clopen neighbourhood of (t1,g1) in G/F which 148 5 Free Products of Pro-C Groups

misses (t2,g2). Since (t1,g1) and (t2,g2) are arbitrary, we have that G/F is totally disconnected.

Let a proﬁnite group G act continuously on a proﬁnite space X. Assume that the canonical epimorphism ϕ : X −→ T = G\X admits a continuous section σ : T −→ X. For each t ∈ T ,letGσ(t) denote the G- stabilizer of σ(t). Then F ={Gσ(t) | t ∈ T } is a family of closed subgroups of G continuously indexed by T (see Lemma 5.2.2 and use the continuity of σ ). Then

Lemma 5.2.4 The map . ψ : G/F = G/Gσ(t) −→ X given by ψ(gGσ(t)) = gσ(t) (g ∈ G, t ∈ T)is a homeomorphism of G-spaces.

Proof The epimorphism ψ1 : T × G −→ X deﬁned by ψ1(t, g) = gσ(t) (g ∈ G, t ∈ T)is continuous since σ is continuous. Clearly ψ1 induces a continuous map of the quotient space ψ : G/F = T × G/∼−→X given by gGσ(t) → gσ(t), which is injective and surjective. Since, according to Proposition 5.2.3, G/F is a proﬁnite space, one deduces that ψ is a homeomorphism.

5.3 Free Pro-C Products: The Internal Viewpoint

Now we want to characterize internally when a profinite group is a free pro-C product of a sheaf of subgroups. To do this we start by giving a definition of an internal free pro-C product. We shall prove eventually that the internal and the external definitions are in fact equivalent. Let G be a pro-C group and let {Gt | t ∈ T } be a family of closed subgroups of G continuously indexed by a profinite space T . We say that G is the internal free pro-C product of this family of subgroups if

(a) Gs ∩ Gt = 1, whenever s,t ∈ T , s = t; and (b) if D = Gt t∈T has the topology induced from G, then the following universal property is sat- isﬁed: whenever β : D −→ H is a continuous map into a pro-C group H such 5.3 Free Pro-C Products: The Internal Viewpoint 149

∈ that its restriction β|Gt to each Gt (t T)is a homomorphism, then there exists a unique extension of β to a continuous homomorphism β¯ : G −→ H . One checks that it sufﬁces to verify the above universal property for groups H ∈ C. C If G is the internal free pro- product! of a continuously indexed family of closed { | ∈ } = C subgroups Gt t T , we write G t∈T Gt ,orsimply,

G = Gt . t∈T ! We shall prove next that if G = ∈ Gt , then there exists a sheaf of pro-C groups ∼ t T G over T such that G(t) = Gt and

G = G. T ! = G G : G −→ And, conversely, if G T , for some sheaf and universal map ω G, then

G = ω G(t) . t∈T In this way we shall establish the equivalence between internal and external free products. Suppose that G is a pro-C group with a continuously indexed family of closed subgroups {Gt | t ∈ T }. Deﬁne an associated sheaf G = (G,π,T)as the subsheaf G = (t, g) ∈ T × G g ∈ Gt of the constant sheaf T × G, where π : G −→ T is the restriction of the natural −1 projection T × G −→ T . Note that G(t) = π (t) ={t}×Gt . To check that G is indeed a sheaf, it remains to show that G is compact, or equivalently that it is closed in T × G; this follows from Lemma 5.2.1. Next deﬁne ω : G −→ D = Gt → G, t∈T by ω(t,g) = g, (g ∈ Gt ,t ∈ T). Then ω is continuous and a morphism since it is the restriction of the projection

T × G −→ G. ! ! = G = Assume now that G t∈T Gt . We claim that T G. To! see this we shall G check that the pair (G, ω) satisfies the universal property of T . Indeed, let H beapro-C group and let β : G −→ H be a morphism. Define β1 : D −→ H by β1(g) = β(t,g),forg ∈ Gt . Then β1 is continuous (if A is closed in H , then −1 = −1 β1 (A) ω(β (A)) is closed in D, since ω is a closed map). Furthermore, β1|Gt 150 5 Free Products of Pro-C Groups ! ∈ = is a homomorphism (t T ). Then by the definition of G t∈T Gt , there exists a unique extension of β1 to a continuous homomorphism

β¯ : G −→ H.

Obviously βω¯ = β, and any other homomorphism from G to H satisfying this last equality coincides with β1 on D. Conversely, assume now that! G = (G,π,T) is a sheaf of pro-C groups over a profinite space T , and let G = G, with universal morphism ω : G −→ G. Define T ∼ Gt = ω(G(t)) (t ∈ T). By Proposition 5.1.6, Gs ∩ Gt = 1, for s = t, and Gt = G(t) (s, t ∈ T). We claim that the family {Gt | t ∈ T } is continuously indexed by T .To seethisweuseLemma5.2.1: first observe that the map G −→ E = (t, g) ∈ T × G g ∈ Gt given by gt ∈ G(t) → (t, ω(gt )) is a continuous surjection, and in fact a homeomorphism; it follows that E is compact,! and so closed in T × G. = Finally, we prove that G t∈T Gt by checking the corresponding universal property: given a pro-C group H and a continuous map β1 : D = Gt −→ H t∈T ∈ = : such that β1|Gt is a homomorphism for each t T , define β β1ω. Then β G −→ H is a morphism. Hence there exists a unique continuous homomorphism ¯ β1 : G −→ H ¯ ¯ such that β1ω = β. It follows that β1 extends β1. Thus we have proved the following lemma. ! C = Lemma 5.3.1 The definitions! of internal free pro- product G t∈T Gt and of C G external free pro- product T are equivalent.

This lemma allows us to use internal or external free pro-C products according to convenience and permits us to translate results from internal to external free pro-C products and vice versa. For example, Proposition 5.1.7 translates, in terms of internal products, to the following result.

Proposition 5.3.2 Let (Ai,ϕij ,I) be an inverse system of pro-C groups over a directed partially ordered set I , and deﬁne = G lim←− Ai. i∈I

= Let (Ti,ρii ,I) be an inverse system over I of proﬁnite! spaces, and deﬁne T lim ∈ T . Suppose that for each i ∈ I , one has A = A , where {A | ←− i I i i τ∈Ti i,τ i,τ 5.3 Free Pro-C Products: The Internal Viewpoint 151

τ ∈ Ti} is a family of closed subgroups of Ai continuously indexed by Ti such that ϕii (Ai,τ ) ≤ Ai ,τ , whenever i, i ∈ I , i i , where τ = ρii (τ). For each t = (τ ) ∈ lim ∈ T = T , deﬁne G = lim ∈ A . Then {G | τ ∈ T } is a family i ←− i I i t ←− i I i,τi t of closed subgroups of G continuously indexed by T , and

G = Gt . t∈T

Corollary 5.3.3 Let G = A B be a free pro-C product of pro-C groups A and B. Denote by N the smallest closed normal subgroup of G containing A. Then

N = b−1Ab. b∈B

Proof (See a different proof in Proposition 5.7.6.) We remark that N is the kernel of the epimorphism G −→ B that sends B identically to B and sends A to 1. Indeed, if K denotes the kernel of this epimorphism, then clearly N ≤ K, G = NB = KB and N ∩ B = K ∩ B = 1; hence an element g ∈ G can be written uniquely as products g = nb = kb (n ∈ N,k ∈ K,b,b ∈ B); thus b = b and n = k;soN = K. Therefore, G/N =∼ B. Assume ﬁrst that B ∈ C. Then N is open in G. In this case the result can be proved directly by verifying it for abstract free products and then taking the pro-C completion. Alternatively, the result follows from the Kurosh subgroup theorem for open subgroups of a free pro-C product (cf. RZ, Theorem 9.1.9). In general, write = = G A B lim←− A (B/U), U∈U where U is the collection of all open normal subgroups of B.LetNU denote the closed normal subgroup generated by A in GU = A (B/U) (U ∈ U). Clearly G = lim ∈U G and N = lim ∈U N . On the other hand, by the case above, ←− U U ←− U U

−1 NU = (bU) A(bU). bU∈B/U

Thus the result follows from Proposition 5.3.2.

The following theorem gives a natural justification for the concept of free pro-C product as defined above; it says that a free pro-C product of pro-C groups indexed by a profinite space is precisely an inverse limit of free pro-C products of finite groups in C indexed by finite spaces. ! = C First we need the following notation. Let G t∈T Gt be the free pro- prod- uct of a continuously indexed family {Gt | t ∈ T } of closed subgroups of a pro-C group G. For a closed subset S of T , denote by G(S) the closed subgroup of G generated by all the Gs , s ∈ S.IfS =∅, then put G(S) = 1. 152 5 Free Products of Pro-C Groups

Theorem 5.3.4 Let G be a pro-C group and let {Gt | t ∈ T } be a family of closed { | ∈ } subgroups indexed by! a proﬁnite space T . Then Gt t T is continuously in- = C dexed by T and G t∈T Gt if and only if there exists a directed partially ordered set I and two inverse systems over I , one (Ai,ϕij ,I)of pro-C groups and the other (Ti,ρij ,I)of ﬁnite discrete spaces such that (a) T = lim ∈ T ; ←− i I i !C (b) for each i ∈ I , Ai = ∈ Ai,τ , where each Ai,τ is a group in C; τ Ti (c) ϕii (Ai,τ ) ≤ Ai ,τ , whenever i i and τ = ρii (τ); (d) for every t = (τ ) ∈ lim ∈ T = T , G = lim ∈ A ; i ←− i I i t ←− i I i,τi (e) G = lim ∈ A . ←− i I i

{ | Proof If conditions (a)Ð(e) are satisﬁed, then! Proposition 5.3.2 implies that Gt ∈ } = C { | t T is continuously indexed and G !t∈T Gt . Conversely, assume that Gt ∈ } = C R t T is continuously indexed and G t∈T Gt .Let be the set of all clopen equivalence relations on T (i.e., those relations on T whose equivalence classes are clopen). Let I = i = (R, U) R ∈ R,Uis an open normal subgroup of G .

Make I into a directed partially ordered set by deﬁning

i = (R, U) i = R ,U to mean that U ≤ U and R ⊆ R (or equivalently that tR ⊆ tR , for all t ∈ T ). For i = (R, U) ∈ I and τ = tR (t ∈ T ), define Ti = T/R, with the quotient space topol- !C ogy, A = G(τ)U/U and A = A (observe that τ = tR is an equivalence i,τ i τ∈Ti i,τ class represented by some t ∈ T , and so τ is a clopen subset of T ). Since each Ti is a finite discrete space and each G(τ)U/U is a group in C, one sees that each Ai is a free pro-C product of finitely many finite groups in C.Ifi = (R, U) i = (R ,U ), define

: = −→ = ρii Ti T/R Ti T/R to be the natural epimorphism of ﬁnite discrete spaces; and deﬁne

ϕii : Ai −→ Ai to be the continuous epimorphism determined by the homomorphisms Ai,τ −→ Ai ,τ , which in turn are induced by the natural inclusion homomorphism G(τ) →

G(τ ). Clearly (Ti,ρij ,I) and (Ai,ϕij ,I) are inverse systems over I , and T = lim ∈ T . ←− i I i Let t = (τ ) ∈ lim ∈ T ∈ T . Note that if i = (R, U) ∈ I , then τ = ρ (t) = tR, i ←− i I i i i where ρi : T −→ Ti is the projection. We claim that = = Gt lim←− Ai,τi lim←− G(τi)U/U. i∈I i∈I 5.4 Proﬁnite G-Spaces vs the Weight w(G) of G 153

To see this note ﬁrst that I = i = R ,U ∈ I G(τi )U /U = G tR U /U = Gt U /U is a directed partially ordered subset of I . Indeed, if i = (R ,U ), i = (R ,U ) ∈ I , put U = U ∩ U , and choose a clopen equivalence relation R on T such that tR ⊆ T(Gt U) and R ⊆ R ∩ R ; then i = (R, U) ∈ I and i i ,i .Next we observe that I is coﬁnal in I ; indeed, if i = (R, U) ∈ I , choose a clopen equivalence relation R on T such that tR ⊆ T(Gt U) and R ⊆ R; then i = (R ,U)∈ I and i i. It follows that

= = lim←− Ai,τi lim←− A(R,U),tR lim←− A(R,U),tR i∈I (R,U)∈I (R,U)∈I = = = lim←− G(tR)U/U lim←− Gt U/U Gt .

(R,U)∈I UoG

This proves the claim. Finally, by Proposition 5.3.2,

lim A = lim A = lim A = G = G. ←− i ←− i,τ ←− i,τ t

5.4 Proﬁnite G-Spaces vs the Weight w(G) of G

The purpose of this section is to show that if a profinite group G acts on a profinite space X, then, under certain conditions, it also acts on a profinite space X˜ that is an epimorphic image of X so that the G-stabilizers of points are preserved, but with the additional property that the topological weight of X˜ is at most the topological weight of G (see Sect. 1.2). This section can be postponed until we deal with the structure of subgroups of free pro-C products (Sect. 9.6), where the weight of subgroups plays a crucial role. Let G be a profinite group and let X be a profinite G-space. For x ∈ X,letGx denote as usual the G-stabilizer of x. By Lemma 5.2.2, {Gx | x ∈ X} is a collection of closed subgroups of G continuously indexed by X. By Lemma 5.2.1, the subset E = (x, g) ∈ X × G g ∈ Gx of the product space X × G is closed, and so E is compact. Let ρ : E −→ G and π : E −→ X be the restrictions of the projections X × G −→ G and X × G −→ X, respectively. Then ρ(E)= D = Gx x∈X 154 5 Free Products of Pro-C Groups is closed in G.Forafixedx ∈ X, define E(x) ={(x, g) | g ∈ Gx}.

ρ E(x) E G

π • X x

−1 Put I ={(x, 1) | x ∈ X, 1 ∈ Gx}. Then I = ρ (1) is a compact subset of E.As- ∩ = ∈ = sume in addition that Gx1 Gx2 1, for x1,x2 X, x1 x2. Note that then the restriction

ρ|(E−I) : E − I −→ D −{1} of ρ to E − I is a bijection (in fact a homeomorphism, as one easily deduces from the argument below). Let U denote the collection of all open normal subgroups U of G. Then

−1 EU = E − ρ (U) is compact (and open in E), for every U ∈ U. Therefore : −→ = − ρ|EU EU ρ(EU ) D U is a homeomorphism. For a given topological space Y , we use the standard notation w(Y) for the minimal cardinality of a base of open sets of the topology of Y (the weight of Y ). Hence w(EU ) ≤ w(G), and since EU is a proﬁnite space, the cardinality of the set of clopen subsets of EU is at most max{w(G),ℵ0} (cf. RZ, Proposition 2.6.1(a)). Since |U|≤max{w(G),ℵ0},wehaveprovedthefollowinglemma.

Lemma 5.4.1 The cardinality of the set

{C | C is a clopen subset of EU , for some U ∈ U} is at most max{w(G),ℵ0}.

Let R denote an equivalence relation on X; as usual, we denote by X/R the space of equivalence classes {xR | x ∈ X} of R, with the quotient topology. We view R as a subset of X × X, or equivalently as a family of disjoint subsets of X (the R-equivalence classes) whose union is X. Recall (see Sects. 2.1 and 2.2) that R is a clopen equivalence relation if each equivalence class xR (x ∈ X) is a clopen subset of X. Since X is compact, the number of equivalence classes of a clopen equivalence relation is finite. For the case we are considering here, i.e., when X is a profinite G-space, we say that R is G-invariant if whenever (x1,x2) ∈ R and g ∈ G, 5.4 Profinite G-Spaces vs the Weight w(G) of G 155 then (gx1,gx2) ∈ R; in other words, if g(xR) = (gx)R, for every x ∈ X, g ∈ G. Hence if R is a G-invariant clopen equivalence relation on X, then X/R is a finite discrete G-space. Consider the subspace X of X defined as

X ={x ∈ X | Gx = 1}.

From now on in this section, we shall assume that X is dense in X: X = X. −1 Since Ggx = gGxg (x ∈ X, g ∈ G), it follows that the action of G on X induces an action of G on X .IfR is a clopen G-invariant equivalence relation on X, then it induces a clopen equivalence relation on X which we denote by R ; observe that R has a ﬁnite number of equivalence classes as well, and in fact

X/R = X /R .

Let R denote the set of all clopen G-invariant equivalence relations R on X.We know that = = X lim←− X/R lim←− X /R R∈R R∈R (see Sect. 1.3), as G-spaces. Observe that if R ∈ R, then the clopen subsets −1 −1 EU ∩ π (xR) = EU ∩ π xR ∩ X x ∈ X determine a clopen equivalence relation on the proﬁnite space EU , for every U ∈ U. It follows from Lemma 5.4.1 that one can choose a subset P of R with |P|≤ max{w(G),ℵ0} and for every R ∈ R, there exists some P ∈ P such that −1 −1 EU ∩ π (xR) x ∈ X = EU ∩ π (xP ) x ∈ X .

In addition, we may assume that P is closed under finite intersections [if P1 and P2 are equivalence relations, its intersection P1 ∩ P2 is the equivalence relation whose equivalence classes are the nonempty intersections of the equivalence classes of P1 and P2], so that X /P P ∈ P is a directed inverse system of finite G-spaces. Define ˜ = X lim←− X /P . P ∈P Then X˜ is a profinite G-space and there exists a natural continuous epimorphism

μ : X −→ X˜ 156 5 Free Products of Pro-C Groups determined by the projections

X −→ X/P = X /P (P ∈ P).

We observe that μ is a morphism of G-spaces, i.e., μ(gx) = gμ(x) (g ∈ G, x ∈ X).

Proposition 5.4.2 Let G be a proﬁnite group and let X be a proﬁnite G-space such ∩ = ∈ = that Gx1 Gx2 1, whenever x1,x2 X, x1 x2. Assume that the subspace

X ={x ∈ X | Gx = 1} of X is dense in X. Then there exists a proﬁnite G-space X˜ and an epimorphism μ : X −→ X˜ of G-spaces such that ˜ (a) w(X) ≤ max{w(G),ℵ0}; ˜ (b) the restriction μ|X : X −→ X of μ to X is an injection; (c) Gx = Gμ(x), for every x ∈ X; (d) for every x ∈ X, the restriction of μ to the orbit Gx of x in X is a homeomorphism onto the orbit Gμ(x) of μ(x) in X˜ .

Proof Construct X˜ as above.

(a) This follows from the fact that |P|≤max{w(G),ℵ0} and each X /P is a ﬁnite discrete space.

(b) Let x1,x2 ∈ X be such that x1 = x2. Choose R ∈ R such that x1R = x2R. = ∈ U ≤ ≤ Since Gx1 ,Gx2 1, there exists some U with Gx1 U and Gx2 U. Then

−1 −1 EU ∩ π (x1R) = EU ∩ π (x2R).

Choose P ∈ P so that −1 −1 EU ∩ π (xR) x ∈ X = EU ∩ π (xP ) x ∈ X .

Then x1P = x2P , and therefore, μ(x1) = μ(x2). (c) Clearly Gx ≤ Gμ(x).IfGx = G, then Gμ(x) = Gx ; otherwise, let g ∈ G−Gx ; then gx = x. Choose R ∈ R so that (gx)R = xR.Let

x1 ∈ (gx)R ∩ X and x2 ∈ xR ∩ X .

Then x1R = x2R. Choose P ∈ P as in the proof of part (b). Then

g x2P = x1P = x2P in X /P .

Therefore gμ(x) = μ(x), and hence g/∈ Gμ(x). Thus Gx = Gμ(x). Part (d) follows from part (c). 5.5 Basic Properties of Free Pro-C Products 157

5.5 Basic Properties of Free Pro-C Products

Throughout this section {Gt | t ∈ T } is a family of closed subgroups of a pro-C group G continuously indexed by a proﬁnite space T . Recall that a collection U of subsets of a set T is ﬁltered from below if whenever U,V ∈ U, there exists some!W ∈ U with W ⊆ U ∩ V . Recall also that if S ⊆c T , { | ∈ } then the closed subgroup of t∈T Gt generated by Gs s S is denoted by G(S). ! = C C Lemma 5.5.1 Assume that G t∈T Gt isafreepro- product of a continuously indexed collection of closed subgroups of a pro-C group G. Let S be a closed subset of T . The following statements hold.

(a) Let {Ui |i ∈ I} be a collection of clopen subsets of T ﬁltered from below such = that S i∈I Ui . Then G(S) = G(Ui). i∈I [See also! Proposition 5.5.3.] = C (b) G(S) t∈S Gt .

Proof We focus on the proof of part (b); part (a) will be proved during the process. Assume first that S is clopen. In this case we prove the result using the external free product notation. Let (G,π,T) be the corresponding sheaf (see the proof of Lemma 5.3.1). Let (GS,πS,S) be the sheaf of pro-C groups over S defined by −1 GS(s) = G(s), s ∈ S. Then GS = π (S), and so GS is a clopen subspace of G.Let ωS : GS −→ G(S) be the restriction of the canonical morphism ω : G −→ G to GS . We! shall show that (G(S), ωS) satisfies the universal property of the free pro-C prod- G : G −→ C uct S S . Indeed, if βS S H is a morphism into a pro- group H , extend it to a morphism β : G −→ H by putting β(G(t)) = 1, for all t ∈ T − S. Then there ¯ ¯ ¯ exists a continuous homomorphism β : G −→ H with βω = β.IfβS is defined to ¯ ¯ be the restriction of β to G(S), then one has that βSωS = βS . The uniqueness of ¯ βS satisfying this property follows from the fact that G(S) is generated by ωS(GS). This proves part (b) when S is clopen. Assume now that S is a closed subset of T .Let{Ui | i ∈ I} be a collection filtered = from below of clopen subsets of T containing S such that S i∈I Ui (for example, the collection of all clopen subsets containing S). Then = = S Ui lim←− Ui, i∈I i∈I

⊆ = !where, as usual, i i in I means that Ui Ui . By the case above, G(Ui)

G . Note that i i implies that G(U ) ≤ G(U ). Using Proposition 5.3.2 t∈Ui t i i one easily sees that

= lim←− G(Ui) Gt . i∈I t∈S 158 5 Free Products of Pro-C Groups

In particular, lim ∈ G(U ) is generated by the G , t ∈ S. On the other hand, since ←− i I i t each G(Ui) is a subset of G, one has = lim←− G(Ui) G(Ui). i∈I i∈I ! So lim ∈ G(U ) = G(S), and thus G(S) = G(U ) = G , proving both ←− i I i i∈I i t∈S t parts (a) and (b). ! = C = Lemma 5.5.2 Assume that G t∈T Gt is a free pro- product. Let T S1 ∪. ···∪. Sn be a decomposition of T as a disjoint union of ﬁnitely many clopen subsets Si . Then G is the free pro-C product

G = G(S1) ··· G(Sn).

Proof Again we use the external free product notation. Let (G,π,T) be the corresponding sheaf (see the proof of Lemma 5.3.1). Let (Gi,πi,Si) be the sheaf over Si −1 deﬁned by Gi = π (Si) and where πi is the restriction of π to Gi (i = 1,...,n). By Lemma 5.5.1,

G(Si) = Gi, Si with canonical map, say, ωi : Gi −→ G(Si). To prove that G is the free pro-C product of the subgroups G(Ui), we shall show that G satisfies the universal property of free products (see Example 5.1.3(a)). Let H beapro-C group and let ϕi : G(Si) −→ H be continuous homomorphisms (i = 1,...,n). Define ψi : Gi −→ H by ψi = ϕiωi , (i = 1,...,n), and let ψ : G −→ H be defined to equal ψi on Gi , (i = 1,...,n). Then clearly ψ is a morphism; it induces a continuous homomorphism ϕ : G −→ H such that ϕω = ψ. Plainly the restriction of ϕ to each G(Si) coincides with ϕi . The uniqueness of ϕ is clear because G = G(Si), . . . , G(Sn). Thus

G = G(S1) ··· G(Sn), as asserted.

The following proposition generalizes Lemma 5.5.1(a). ! = C Proposition 5.5.3 Let G t∈T Gt beafreepro- product. Then for every family {Si | i ∈ I} of closed subsets of T , G(Si) = G Si . i∈I i∈I

Proof Assume ﬁrst that I is ﬁnite and that all Si,i ∈ I , are clopen subsets of T . By induction one may assume that I consists of two elements I ={1, 2}.Using 5.5 Basic Properties of Free Pro-C Products 159

Lemma 5.5.2 one has

G(S1 ∪ S2) = G(S1 − S2) G(S1 ∩ S2) G(S2 − S1),

G(S1) = G(S1 − S2) G(S1 ∩ S2), and

G(S2) = G(S1 ∩ S2) G(S2 − S1).

Let ϕ be the continuous endomorphism of G(S1 ∪ S2) which is the identity on the free factors G(S1 −S2) and G(S1 ∩S2) and maps G(S2 −S1) to the identity element. Then G(S1) is sent identically to itself and, in particular, if g ∈ G(S1) ∩ G(S2), then ϕ(g) = g; on the other hand, since ϕ(G(S2)) ≤ G(S1 ∩ S2), we deduce that g = ϕ(g) ∈ G(S1 ∩ S2). Thus G(S1) ∩ G(S2) ⊆ G(S1 ∩ S2). The reverse inclusion G(S1) ∩ G(S2) ⊇ G(S1 ∩ S2) is obvious. U We return now to the general case. Let i be the collection of all clopen subsets of T containing S (i ∈ I); hence S = U. Deﬁne U = U .By i i U∈Ui i∈I i Lemma 5.5.1(a) G(S ) = G(U). Denote by W the collection of all clopen i U∈Ui subsets W of T of the form W = U1 ∩···∩Un, where Uj ∈ U,forj = 1,...,n and n = 1, 2,.... Then W is a collection of clopen subsets of T ﬁltered from be- = low. Therefore, by Lemma 5.5.1(a), G( W∈W W) W∈W G(W). Hence, since = clearly W∈W W i∈I Si , we obtain that G Si = G(W). i∈I W∈W

Finally, note that if W = U1 ∩···∩Un, then G(W) = G(U1) ∩···∩G(Un),as shown above. Thus G Si = G(W) = G(U) = G(U) = G(Si). i∈I W∈W U∈U i∈I U∈Ui i∈I

Next we extend Lemma 5.5.2. ! = C Proposition 5.5.4 Let G t∈T Gt be a free pro- product and let

ρ : T −→ X be a continuous surjective map of proﬁnite spaces. For each x ∈ X, put Sx = ρ−1(x). Then

G = G(Sx). x∈X

Proof Write X = lim ∈ X , where X ranges through the ﬁnite quotient spaces ←− i I i i of X.Letϕi : X −→ Xi and ϕij : Xi −→ Xj denote the corresponding surjections 160 5 Free Products of Pro-C Groups

∈ ∈ = ∈ = −1 (i, j I,i j). For each i I deﬁne ρi ϕiρ.Forx Xi , deﬁne Ri(x) ρi (x). Then, by Lemma 5.5.2,

G = G Ri(x) . x∈Xi

Note that if i j and ϕij (xi) = xj ,forsomexi ∈ Xi and xj ∈ Xj , then Ri(xi) ⊆ Rj (xj ). So we can apply Proposition 5.3.2, letting G play the role of Ai , for each i, and G(Ri(x)) play the role of Ai,τ , to get

= = G lim←− G Ri(xi) lim←− G Ri(xi) ∈ ∈ i I Xi X i I = G Ri(xi) = G(Sx), x∈X i∈I x∈X where x = (x ) ∈ lim ∈ X = X. i ←− i I i

In Theorem 5.5.6 we study the subgroup of a free pro-C product generated by subgroups of the free factors. We ﬁrst need an auxiliary result. ! = C Lemma 5.5.5 Let G r∈T Gr beafreepro- product. Suppose that for each ∈ r T , Hr is a closed subgroup of Gr such that r∈T Hr is a closed subset of G. Let H be the closed subgroup of G generated by the subgroups Hr . Then H ∩ Gt = Ht , for every t ∈ T .

Proof Fix t ∈ T .IfHt = Gt , the result is trivial; so we shall assume Ht

Then H is closed in G, and so H is a sheaf over T whose s-ﬁber H(s) is isomorphic with Hs (s ∈ T)(see Proposition 5.1.6(c)). Let z ∈ G(t) with ω(z) = gt ; then z/∈ H(t).LetU be an open normal subgroup of G(t) such that z/∈ UH(t).De- ﬁne K = G(t)/U.Ifσt : G(t) −→ K is the canonical epimorphism, then σt (z)∈ / σ(H(t)) = B. By Corollary 5.1.5, σt extends to a morphism σ : G −→ K.Let S = s ∈ T H(s) ≤ σ −1(B) .

Then t ∈ S and T − S = π(H − σ −1(B)). Hence S is open in T .LetV be a clopen subset of T such that t ∈ V ⊆ S. Deﬁne a morphism β : G −→ K by putting β(x) = σ(x),forx ∈ π −1(V ), and β(x) = 1, for x ∈ π −1(T − V).Letβ¯ : G −→ K be the 5.5 Basic Properties of Free Pro-C Products 161

¯ ¯ continuous homomorphism such that βω = β. Then β(Hs) ≤ B, for all s ∈ T ,so ¯ ¯ that β(H) ≤ B. On the other hand, β(gt )/∈ B. Therefore gt ∈/ H , as asserted.

Theorem 5.5.6 Let C be either an extension-closed! pseudovariety of ﬁnite groups = C or a pseudovariety of ﬁnite abelian groups. Let G t∈T Gt be a free pro- prod- C ∈ uct of pro- groups Gt . For each t T , let Ht be a closed subgroup of Gt , and assume that t∈T Ht is a closed subset of G. Then the closed subgroup H of G generated by the subgroups Ht , t ∈ T , is the free pro-C product

H = Ht . t∈T

Proof Assume first that T is a finite indexing set. Then, when C is extension-closed, the result is proved in Corollary 9.1.7 of RZ; and when C is a pseudovariety! of finite abelian groups, the result is obvious, since in this case the free product t∈T Gt is simply the direct sum of the groups Gt as abelian pro-C groups. Consider now the general case. First use Theorem 5.3.4 to express G!as an inverse limit G = lim ∈ A , where each A is a free pro-C product A = A ←− i I i i i τ∈Ti i,τ of groups in C and the spaces Ti are finite. Let ϕi : G −→ Ai be the canonical projection. Note that this decomposition is constructed in such a way that for any given t ∈ T and any i ∈ I , there exists some τ ∈ Ti such that ϕi(Gt ) ≤ Ai,τ .LetHi,τ be the closed subgroup of Ai,τ generated by all the subgroups ϕi(Gt )(t∈ T)such that ϕi(Gt ) ≤ Ai,τ .LetHi = ϕi(H ). Then Hi is the closed subgroup of Ai generated by the Hi,τ , τ ∈ Ti . Hence, by the case considered above,

Hi = Hi,τ . τ∈Ti

It then follows from Theorem 5.3.4 that

˜ H = Ht , t∈T

˜ with H = lim ∈ H , where for each i ∈ I , τ is the projection of t on T . Finally t ←− i I i,τ i ˜ ˜ observe that for each t ∈ T one clearly has Ht ≥ Ht ; and on the other hand Ht ≤ ˜ H ∩ Gt = Ht , by Lemma 5.5.5. Thus Ht = Ht .

Next we study the normal subgroup of a free pro-C product generated by normal subgroups of the factors. ! = C Lemma 5.5.7 Let G r∈T Gr beafreepro- product . Suppose that for each ∈ r T , Nr is a closed normal subgroup of Gr such that r∈T Nr is a closed subset of G. 162 5 Free Products of Pro-C Groups

(a) Fix t = s in T and let gs ∈ Gs − Ns . Then there exists a ﬁnite quotient group K of Gs/Ns and a continuous homomorphism ψ : G → K extending the canonical epimorphism Gs −→ K such that ψ(gs) = 1, ψ(Gt ) = 1 and Nr ≤ Ker(ψ), for all r ∈ T . (b) Let N be the smallest closed normal subgroup of G containing Nr , for every r ∈ T . Then

(b1) Gr ∩ N = Nr , for every r ∈ T ; and (b2) if t = s are in T , then Gt N ∩ GsN = N.

Proof (a) The argument is very similar to the one used in Lemma 5.5.5! .Let G C = ( ,π,T)be the sheaf over T associated with the free pro- product G r∈T Gr , and denote by ω : G −→ G the canonical morphism. Deﬁne −1 H = ω Nr . r∈T

Then H is closed in G, and so H is a sheaf over T whose fiber over r ∈ T is Nr .Let W be an open normal subgroup of Gs containing Ns such that gs ∈/ W . Define K = Gs/W and let σs : Gs −→ K be the canonical epimorphism. By Corollary 5.1.5, there exists a morphism σ : G −→ K extending σs such that σ(G(t)) = 1. Since K is finite, σ −1(1) is open in G. Put − S = r ∈ T H(r) ⊆ σ 1(1) .

Then T − S = π(H − σ −1(1));soT − S is compact, and hence S is open in T . Therefore, there exists a clopen subset V of T such that s ∈ V ⊆ S. It follows that π −1(V ) is a clopen subset of G. Deﬁne β : G −→ K by β = σ on π −1(V ), and β(G(r)) = 1ifr ∈ T − V ; then β is a morphism. Let ψ = β¯ : G −→ K be the continuous homomorphism induced by β so that ψω = β. Then ψ satisﬁes the required conditions. (b) Obviously Gs ∩ N ≥ Ns . If there exists a gs ∈ Gs ∩ N − Ns , consider a morphism ψ : G −→ K as in part (a). Then N ≤ Ker(ψ) and ψ(gv) = 1; therefore, gs ∈/ N, a contradiction. Thus Gs ∩ N = Ns , proving part (b1). Next, let x ∈ Gt N ∩ GsN − N. Then there exists a gs ∈ Gs − N such that x = gsn,forsomen ∈ N. Consider a morphism ψ : G −→ K as in part (a). Since ψ(Gt N) = 1, we have ψ(x) = 1, and so ψ(gs) = 1, contradicting (a). Thus Gt N ∩ GsN = N, proving (b2). ! = C Proposition 5.5.8 Let G r∈T Gr beafreepro- product. Suppose that for ∈ each r T , Nr is a closed normal subgroup of Gr such that r∈T Nr is a closed subset of G. Let N be the closed normal subgroup generated by the Nr (r ∈ T). Then G/N is the free pro-C product

G/N = (Gr /Nr ). r∈T 5.5 Basic Properties of Free Pro-C Products 163

Proof We shall actually show that the collection {Gr N/N | r ∈ T } of closed subgroups of G/N is continuously indexed by T and

G/N = (Gr N/N) r∈T (free pro-C product). Then the result will follow since according to Lemma 5.5.7, ∼ N ∩ Gr = Nr , so that Gr N/N = Gr /Nr . To check that the collection {Gr N/N | r ∈ T } is continuously indexed by T , it remains to show that if U is an open neighbourhood of 1 in G/N, then {r ∈ T | Gr N/N ≤ U} is open in T ; but this is clear from the assumption that the collection {Gr | r ∈ T } of closed subgroups of G is continuously indexed by T . Finally, one easily shows that the required universal property

G/N H

r∈T (Gr N/N) ! = follows immediately from the corresponding universal property for G r∈T Gr , and the fact that G/N = Gr N/N | r ∈ T . Thus

G/N = (Gr N/N), r∈T as desired. ! = C Corollary 5.5.9 Let G r∈T Gr beafreepro-! product and let N beaclosed = C normal subgroup of G. Then G/N r∈T Gr N/N (free pro- product) if and only if N is generated by the intersections N ∩ Gr (r ∈ T)as a closed normal subgroup of G.

Proof Suppose N is generated by the intersections Nr = N ∩Gr (r ∈ T)as a closed normal subgroup of G. We observe that Nr = Gr ∩ N, r∈T r∈T = !and so r∈T Nr is a closed subspace of G. Therefore, by Proposition 5.5.8, G/N ∈ Gr N/N. ! r T Conversely, assume G/N = ∈ Gr N/N.LetN be the closed normal sub- r T group of G generated! by the subgroups N ∩ Gr (r ∈ T). Then N ≤ N. By Proposi- = tion 5.5.8, G/N r∈T Gr N /N . Consider the natural continuous epimorphism

ψ : G/N = Gr N /N −→ G/N = Gr N/N. r∈T r∈T 164 5 Free Products of Pro-C Groups

Note that Ker(ψ) = N/N and that (N/N )∩(Gr N /N ) = 1inG/N ; i.e., ψ sends

Gr N /N isomorphically to Gr N/N. Since r∈T (Gr N /N ) and r∈T (Gr N/N) are compact spaces, it follows that the map

ψ : Gr N /N −→ (Gr N/N) r∈T r∈T induced by ψ is a homeomorphism. Hence ψ −1 induces a continuous homomorphism

ϕ : G/N = Gr N/N −→ G/N = Gr N /N r∈T r∈T which is the inverse of ψ. Thus ψ is an isomorphism, and so N = N .

Proposition 5.5.10 Let C be a pseudovariety of ﬁnite groups closed under extensions with abelian kernel. Let {Gx | x ∈ X} be a family of closed subgroups of a pro-C group G continuously indexed by a proﬁnite space X, and assume that G is the free pro-C product of this family:

G = Gx. x∈X = = Assume in addition that Gx1 1 Gx2 , where x1,x2 are two different points of X. Then G is not abelian.

Proof Let U1 be a clopen neighbourhood of x1 in X such that x2 ∈/ U1. Deﬁne U2 = X − U1. Then G = G(U1) G(U2), according to Lemma 5.5.2. Hence we may assume that X ={x1,x2}, and G = G1 G2.LetV be an open normal subgroup of G such that G1 ≤ V and G2 ≤ V . Then there exists a continuous epimorphism

G −→ (G1/G1 ∩ V) (G2/G2 ∩ V), and hence we may assume that G1 and G2 are ﬁnite and abelian. Finally, since C is closed under extensions with abelian kernel, the wreath product G1 G2 = (G1 ×···×G2) G2 is in C. From the universal property of free products we deduce that there is a continuous epimorphism G −→ G1 G2. Since G1 G2 is not abelian, neither is G.

5.6 Free Products and Change of Pseudovariety

Free products depend on the pseudovariety C over which they are deﬁned. The next theorem indicates how free products change when we change the pseudovariety C. If H is a proﬁnite group, let RC(H ) = {N | N o H, H/N ∈ C}.

Then H/RC(H ) is the maximal pro-C quotient of H . 5.6 Free Products and Change of Pseudovariety 165

C ⊆ C = Theorem! 5.6.1 Let be pseudovarieties of ﬁnite groups. Assume that G C C C t∈T Gt isafreepro- product of pro- groups Gt . Then C G/RC (G) = Gt /RC (Gt ) . t∈T

Before proving this result we need the following general facts.

Lemma 5.6.2 Let {Gi,ϕij ,I} be an inverse system of proﬁnite groups over a poset I and let = G lim←− Gi. i∈I Then for any pseudovariety of ﬁnite groups C one has = RC(G) lim←− RC(Gi). i∈I

Proof Let ϕi : G −→ Gi be the canonical projection (i ∈ I). Note that ϕi(RC(G)) ≤ RC(Gi), for each i ∈ I .So

= ≤ RC(G) lim←− ϕi RC(G) lim←− RC(Gi). i∈I i∈I

Now, assume that x ∈ G − RC(G); then there exists an L ∈ C and an epimorphism ρ : G −→ L such that x/∈ Ker(ρ). We must show that there exists some k ∈ I such that ϕ (x)∈ / RC(G ); this will imply that x/∈ lim ∈ RC(G ). Since L is ﬁnite, there k k ←− i I i exists some k ∈ I and a continuous homomorphism ρk : Gk −→ L such that ρ = ρkϕk (see Sect. 1.3). Hence ϕk(x)∈ / Ker(ρk); thus ϕk(x)∈ / RC(Gk), as required.

Corollary 5.6.3 Let {Gi | i ∈ I} be a collection, filtered from below, of closed subgroups of a profinite group G, and let C be a pseudovariety of finite groups. Then RC Gi = RC(Gi). i∈I i∈I

Proof of Theorem 5.6.1 Let S be a clopen subset of T . We claim that RC (G(S)) = G(S) ∩ RC (G). To see this note that G = G(S) G(T − S), according to Lemma 5.5.2.Letϕ : G −→ G(S) be the canonical projection. Then

RC G(S) ≤ G(S) ∩ RC (G) = ϕ G(S) ∩ RC (G) ≤ ϕ RC (G) ≤ RC G(S)

(for the last containment relation see RZ, Lemma 3.4.1(c)), proving the claim. For t ∈ T ,letSt be the collection of all clopen subsets of T containing t. Then, using Corollary 5.6.3, Proposition 5.5.3 and the claim above, we have RC (Gt ) = RC G(S) = G(S) ∩ RC (G) = Gt ∩ RC (G).

S∈St S∈St 166 5 Free Products of Pro-C Groups

Hence RC (Gt ) = Gt ∩ RC (G), t∈T t∈T and therefore t∈T RC (Gt ) is closed in G. Let N be the smallest closed normal subgroup of G containing the RC (Gt ) (t ∈ T). Then, according to Proposition 5.5.8, C C G/N = Gt N/N = Gt /RC (Gt ) . t∈T t∈T

Now, to prove that G/RC (G) is the free pro-C product of the groups Gt RC (G)/ RC (G) (= Gt /RC (Gt )) (t ∈ T), we prove ﬁrst that

Gr RC (G)/RC (G) ∩ GsRC (G)/RC (G) is trivial whenever r, s ∈ T and r = s. Indeed, let gs ∈ Gs − RC (G). We need to ∈ prove that gsRC (G) / G!r RC (G)/RC (G). For this we ﬁrst apply Lemma 5.5.7 to = C the free product G/N t∈T Gt N/N to obtain a ﬁnite quotient L of GsN/N and a continuous homomorphism ψ : G/N −→ L extending GsN/N −→ L such that ∼ ψ(gsN)= 1 and ψ(Gr N/N)= 1. Since GsN/N = Gs/(N ∩ Gs) = Gs/RC (Gs), ψ we deduce that L ∈ C . Therefore the homomorphism G −→ G/N −→ L induces a homomorphism ψ : G/RC (G) −→ L. Denote by

ρ : G/N −→ G/RC (G) the canonical epimorphism. Then ρ maps Gt N/N (isomorphically) onto Gt RC (G)/

RC (G) (t ∈ T). Moreover, ψ ρ = ψ. Hence ψ (gsRC (G)) = 1 and ψ (Gr RC (G)/ RC (G)) = 1. Therefore gsRC (G)∈ / Gr RC (G)/RC (G), as needed. Finally, we verify the corresponding universal property. Let K ∈ C , and let β : Gt RC (G)/RC (G) −→ K t∈T be a continuous map whose restriction to each Gt RC (G)/RC (G) (t ∈ T)is a homomorphism. Let β be the composition β Gt −→ Gt RC (G)/RC (G) −→ K. t∈T t∈T ! = C By the universal property of G t∈T Gt , there exists a unique continuous homo- : −→ ∈ C morphism β G K such that β restricts to β on t∈T Gt . Since K , ¯ the homomorphism β factors through a unique continuous homomorphism β : −→ ¯ G/RC (G) K. One deduces that β restricts to β on t∈T (Gt RC (G)/RC (G)). The uniqueness of β¯ follows from the fact that G/RC (G) is generated by the Gt RC (G)/RC (G) (t ∈ T). 5.7 Constant and Pseudoconstant Sheaves 167

We end this section by establishing some notation. Let A be a pseudovariety of ﬁnite abelian groups, and let (G,π,T)be a sheaf of pro-A groups. Let A be its free pro-A product. Note that, in particular, A and each ﬁber G(t) are abelian groups. Let At be the canonical image of G(t) in A(t∈ T). Then we usually write

"A "A " " A = At or A = G, or simply A = At or A = G, t∈T T t∈T T ! ! = A = A G rather than A t∈T At or A t∈T . We refer to this as a direct sum indexed by a topological space, rather than a ‘free proabelian product’.

Example 5.6.4 #A # (a) If T is a finite space, then t∈T At is just the usual direct sum t∈T At endowed with the product topology. (b) Let T be a profinite space and# let A be the class of all finite abelian groups. If =∼ ∈ At Z, for each t T , then t∈T At is just the free abelian profinite group ∼ ∈ with# basis T , while if p is a fixed prime number and At = Zp, for each t T , then t∈T At is just the free abelian pro-p group with basis T . . (c) Let T = S ∪ {∗}be the one-point compactification of a discrete space S.As- G = . sume that t∈T At is a sheaf of groups over S converging to 1 (see Exam- ple 5.1.1(c)). Then "A At = As, t∈T s∈S

the Cartesian product of the groups As with the product topology.

5.7 Constant and Pseudoconstant Sheaves

Throughout this section we assume that the pseudovariety of ﬁnite groups C is extension-closed.

Let (X, ∗) be a pointed profinite space and let A beapro-C group. Define the pseudoconstant sheaf K = K(X,∗)(A) over (X, ∗) to be a sheaf whose fibers are the groups A, if x ∈ X −{∗}; K(x) =∼ {∗} = {1}, if x =∗, and where the topology of K(X,∗)(A) is given as follows: we identify the subspace K(X,∗)(A)−{∗}with (X −{∗})×A endowed with the product topology; and a basis ∗ K for the open neighbourhoods of in (X,∗)(A) consists of the subsets of the form . K ∗ x∈U (x), where U is an open neighbourhood of in X. Note that this defines 168 5 Free Products of Pro-C Groups indeed a sheaf over X, i.e., that the above definition makes K(X,∗)(A) into a profinite space, and moreover the map π : K(X,∗)(A) −→ X that sends Ax to x is continuous. Let

K = K(X,∗)(A) = K(X,∗)(A) = Ax x∈X be the corresponding free pro-C product, where A∗ = w({∗}) ={1}, Ax = ω(K(x)) = ω({x}×A),ifx ∈ X −{∗}, and where

ω : K(X,∗)(A) −→ K is the canonical morphism (see Sect. 5.1). For a ∈ A and x ∈ X −{∗}, we write ax = ω(x,a), and we put a∗ = 1. For example, if A = ZCˆ (the free pro-C group of rank 1), then K(X,∗)(A) is just the free pro-C group on the pointed profinite space (X, ∗). Observe that if ρ : (X, ∗) −→ (Y, ∗) is a map of pointed profinite spaces, it induces a morphism of sheaves K(X,∗)(A) −→ K(Y,∗)(A) which sends ∗ to ∗, and on K(X,∗)(A) −{∗}it is defined as (ρ(x), a), if ρ(x)= ∗; (x, a) → ∗, if ρ(x)=∗.

And this morphism induces in turn a unique continuous homomorphism

ρ˜ : K(X,∗)(A) −→ K(Y,∗)(A) such that ρ(a˜ x) = aρ(x) (x ∈ X). Moreover, the correspondence ρ →ρ ˜ is functorial: $ = ∗ −→ρ ∗ −→η ∗ idX idK(X,∗)(A), and if (X, ) (Y, ) (Z, ) are maps of proﬁnite pointed spaces, then ηρ% =˜ηρ˜.

Remark 5.7.1 Given a profinite space X we can consider a corresponding pointed profinite space (X ∪. {∗}, ∗), with ∗ an isolated point. The associated pseudoconstant sheaf K(X∪. {∗},∗)(A) over this pointed space with fiber A can be identified with (X × A) ∪. {∗}. Observe that a morphism whose domain is this sheaf is determined by a morphism on the constant sheaf X × A over X with fiber A, which we denote by KX(A). Note that

K(X∪. {∗},∗)(A) = KX(A) = Ax, x∈X ∼ where Ax = A, for all x ∈ X. Most results in this section will be stated for pseudoconstant sheaves and corresponding free products over pointed proﬁnite spaces, but they have an obvious counterpart for constant sheaves and corresponding free products over proﬁnite spaces. 5.7 Constant and Pseudoconstant Sheaves 169

Let R be the set of open equivalence relations on X.ForaﬁxedR ∈ R, denote by ρR : (X, ∗) −→ (X/R, ∗R) the quotient map, and let

ρ˜R : K(X,∗)(A) −→ K(X/R,∗R)(A) denote the corresponding continuous homomorphism. If R,R ∈ R and R R ,let ρ˜R ,R : K(X/R ,∗R )(A) −→ K(X/R,∗R)(A) denote the natural continuous homomorphism. Then {K(X/R,∗R)(A), ρ˜R ,R} is an inverse system of pro-C groups, and we have the following result.

Proposition 5.7.2 = = K K(X,∗)(A) lim←− K(X/R,∗R)(A), R∈R where R is the collection of all open equivalence relations on X.

Proof Note that K ∗ (A) = lim ∈RK ∗ (A). So the result is a special case (X, ) ←− R (X/R, R) of Proposition 5.1.7.

Let L beapro-C group and assume that (X, ∗) is a pointed profinite L-space, i.e., that the profinite group L acts continuously on the space X fixing ∗. Then each r ∈ L determines an automorphism of sheaves

K(X,∗)(A) −→ K(X,∗)(A) that sends (x, a) to (xr, a) (x ∈ X −{∗},a∈ A) and ﬁxes! ∗. This in turn induces a r unique continuous automorphism of K = K(X,∗)(A) = K(X,∗)(A), denoted (−) r = ∈ which sends ax to ax axr (a A). Deﬁne an action of L on K

η = ηX : K × L −→ K

r by ηX(k, r) = k (k ∈ K,r ∈ L).

Lemma 5.7.3 (a) Let (Y, ∗) be another proﬁnite L-space and let ρ : (X, ∗) −→ (Y, ∗) be an L- map of proﬁnite pointed spaces, i.e., ρ(xr) = ρ(x)r (x ∈ X, r ∈ L). Then the diagram

ηX K(X,∗)(A) × L K(X,∗)(A)

ρ˜×idL ρ˜

K(Y,∗)(A) × L K(Y,∗)(A) ηY

commutes, i.e., the homomorphisms ρ˜ is compatible with the action of L. 170 5 Free Products of Pro-C Groups

(b) The map η = ηX is continuous; in other words, the action of L on K = K(X,∗)(A) is continuous.

Proof Part (a) follows immediately from the definitions. Part (b) is clear if L is finite, for in this case the statement is equivalent to the continuity of (−)r for a fixed r ∈ L; assume then that L is a general profinite group. Consider first the case where X is finite. Express the action of L on (X, ∗) as a continuous homomorphism α : L −→ Homeo(X, ∗) (cf. Remark 5.6.1 in RZ). Since Homeo(X, ∗) is finite, this homomorphism factors through a finite quotient group L˜ of L, i.e., there exists a continuous homomorphism α˜ : L˜ −→ Homeo(X, ∗) such that α is the composition

α˜ L −→ L˜ −→ Homeo(X, ∗)

(see Sect. 1.3). Since L˜ is ﬁnite, the corresponding map η˜ : K × L˜ −→ K is continuous. From the commutative diagram

η K × L K

η˜ K × L˜ it follows that η is continuous. Consider now a general X. Then (see Sect. 1.3) there is a decomposition ∗ = ∗ (X, ) lim←− (Xi, ) i∈I ! ∗ = K as L-spaces, where each (Xi, ) is a ﬁnite pointed L-space. Put Ki (Xi ,∗)(A). Since each Xi is ﬁnite, the corresponding map

: × −→ → r ∈ ∈ ηi Ki L Ki (ki,r) ki ,ki Ki,r L , is continuous by the case above. By Proposition 5.7.2, = K lim←− Ki. i∈I

Thus η = lim ∈ η is continuous. ←− i I i

It follows from part (b) of this lemma that the corresponding semidirect product K L = K(X,∗)(A) L is a proﬁnite group; and, a fortiori, a pro-C group since C is extension-closed. We collect the above information in the following proposition.

Proposition 5.7.4! Let L be a pro-C group and let (X, ∗) be a pointed proﬁnite L- space. Let K = K(X,∗)(A) be the free pro-C product of a pseudoconstant sheaf 5.7 Constant and Pseudoconstant Sheaves 171

K(X,∗)(A) over (X, ∗) with ﬁber a pro-C group A. Then there exists a natural con- r = ∈ ∈ ∈ tinuous action of L on K determined by ax axr (x X, r L,a A), and hence a semidirect product

K L = K(X,∗)(A) L which is a pro-C group.

If R is an L-invariant open equivalence relation on X, then (X/R, ∗R) is a ﬁnite pointed L-space and the projection (X, ∗) −→ (X/R, ∗R) is a continuous L-map; this map induces a continuous epimorphism

K(X,∗)(A) L −→ K(X/R,∗R)(A) L.

Observe that one can choose a collection R of L-invariant open equivalence relations on X such that ∗ = ∗ (X, ) lim←− (X/R, R) R∈R

(see Sect. 1.3). Then {K(X/R,∗)(A) L | R ∈ R} is an inverse system of pro-C groups, and we deduce

Corollary 5.7.5

= K(X,∗)(A) L lim←− K(X/R,∗R)(A) L . R∈R

The ﬁrst part of the following proposition provides a different and more direct proof of Corollary 5.3.3.

Proposition 5.7.6 Let A and B be pro-C groups and let G = A B be their free pro-C product. Let K denote the closed normal subgroup of G generated by A. Then (a) There exists an isomorphism of pro-C groups

K −→ KB (A) = Ab, b∈B ! b −1 that sends A = b Ab to Ab = ω({b}×A), where ω : KB (A) −→ KB (A) is the canonical morphism. (b) ∼ G = K B = Ab B, b∈B where the action of B on K is determined by the natural action of B on B via

right multiplication: b ∈ B sends Ab to Abb . 172 5 Free Products of Pro-C Groups

Proof Observe that K ∩ B = 1, since if σ : G −→ A × B is the homomorphism that sends A and B identically to A and B, respectively, then σ(K) = A. Hence = G K B. The! action of B on B by right multiplication induces a continuous action of B on b∈B Ab according to Lemma 5.7.3. Hence we have a well-defined semidirect product ˜ G = Ab B, b∈B which is a pro-C group since C is extension-closed. Define continuous homomorphisms ϕ ˜ G = A B = K B G = Ab B ψ b∈B as follows. The subgroup B of G is sent by ϕ to the subgroup B of G˜ identically; ˜ and ϕ sends the subgroup A of G to the subgroup A1 of G identically. Consider the K −→ → b = −1 ∈ ≤ ∈ morphism B (A) G of sheaves given by (b, a) a ! b ab K G(a ∈ : −→ ≤ A,b! B); this induces a continuous homomorphism ψ b∈B Ab K G.If ∈ ∈ = : ˜ −→ x b∈B Ab and b B, define ψ(xb ) ψ(x)b ; then obviously ψ G G is continuous, and one checks that it is a homomorphism: it suffices to observe that ψ = = is compatible with the action of B. Finally,! observe that ψϕ idG and ϕψ! idG˜ . = =∼ Thus ψ and ϕ are isomorphisms. Since ψ( b∈B Ab) K,wehaveK b∈B Ab, proving parts (a) and (b).

We have the following converse to Proposition 5.7.6. Recall that if a proﬁnite group B acts freely on a proﬁnite space X, then the projection π : X −→ X/B admits a continuous section (see Sect. 1.3).

Proposition 5.7.7 Let a pro-C group B act freely on a profinite space X on the right. Let π : X −→ X/B be the projection map, and let σ : X/B −→ X be a continuous section of π. Put Y = σ(X/B). Let K = KX(A) be a constant sheaf over X with fiber the pro-C group A, and let G = K B = Ax B x∈X ! be the pro-C group defined as in Proposition 5.7.4 (Ax = A,∀x ∈ X), where denotes the ‘free pro-C product’. Then there exists an isomorphism of profinite groups ∼ G = R = Ay B. y∈Y 5.7 Constant and Pseudoconstant Sheaves 173

Proof Observe that Y = σ(X/B) is a closed subspace of X, and hence the sheaf KY (A) is a subsheaf of KX(A). Define continuous homomorphisms ψ G = Ax B R = Ay B ϕ x∈X y∈Y ! as follows. The homomorphism ϕ sends! B identically to B, and it sends y∈Y Ay to its natural copy as a subgroup of y∈X Ax . To define the map ψ, we first define a function β : X −→ B by the formula x = yβ(x) where y = σπ(x) (x ∈ X).Note that β(x) exists and it is unique since B acts freely on X. Furthermore, β is continuous since X is homeomorphic with Y ×B as B-spaces (cf. RZ, Corollary 5.6.6), and pr β is the composition of this homeomorphism with the projection: X ≈ Y ×B −→B B. Next we define

ψ : Ax −→ Ay ≤ R x∈X y∈Y to be the continuous homomorphism induced by the sheaf map KX(A) = X × A −→ R given by

= β(x) ∈ ∈ = ψ(x, a) ay x X, a A,y σπ(x) . ! Define the restriction of ψ to x∈X Ax to be ψ, and let ψ send B to B identically. To verify that this indeed defines a homomorphism ψ, we must check that ψ is compatible with the action of B. Observe first that xb = σ π(x)β(x)b = σπ(xb)β(x)b; hence β(xb) = β(x)b.So

b = = β(x)b = b ψ (x, a) ψ(xb, a) ay ψ(x, a) , as required. Finally, note that ψϕ and ϕψ are identity maps. So ψ and ϕ are isomorphisms.

Corollary 5.7.8 Let X be a profinite space and consider the profinite space X˜ = X ∪. {∗}, where the point ∗ is isolated. Assume that a pro-C group B acts freely on the pointed space (X,˜ ∗), i.e., B fixes ∗ and acts freely on X. Consider the sheaf K˜ ˜ over X defined as follows: the subsheaf over X is the constant sheaf KX(A) over X with fiber a pro-C group A, and the fiber over ∗ is an arbitrary pro-C group A∗ (the topology of KX(A) ∪. ({∗} × A∗) is the disjoint topology). Define ˜ G = K B = A A∗ B X˜ X analogously to the definition in Proposition 5.7.4. Then ∼ G = A (A∗ × B). X/B 174 5 Free Products of Pro-C Groups

Proof Note that B centralizes A∗ since B ﬁxes ∗. Therefore the result follows from the proposition above.

Lemma 5.7.9 Let G be a proﬁnite group, and let A and B be closed subsets of G. Then T = g ∈ G Ag ⊆ B is a closed subset of G.

g Proof For a ∈ A consider the continuous map ρa : G −→ G given by ρa(g) = a (g ∈ G). Then = −1 T ρa (B). a∈A

Since ρa is continuous and B is closed, the result follows.

The next two results are included in this section for completeness, although their proofs need results that will be stated only in Chap. 7.

Lemma 5.7.10 Let p be a fixed prime number and let C be a cyclic group of order p. Let (X, ∗) be a pointed profinite space on which C acts freely. Consider the constant sheaf K = KX(A) over the profinite space X with fiber a pro-p group A, and let

K = K = Ax X x∈X ∼ be the free pro-p product of this sheaf (Ax = A, ∀x ∈ X). Let G = K C = K C X be the corresponding semidirect product.

(a) Put L = A∗,C. Then L = A∗ × C. ∼ 2 (b) Assume that A = Cp. Then every ﬁnite subgroup of G has order at most p . ∼ (c) Assume that A = Cp. Then if M is a subgroup of G isomorphic to Cp × Cp, there exists a k ∈ K such that Lk = M. (d) Assume A is torsion-free. Then every ﬁnite subgroup of G is conjugate to a subgroup of C.

Proof Part (a) is clear since C centralizes A∗. Since C is ﬁnite and acts freely on the pointed space (X, ∗), we can express (X, ∗) as an inverse limit X = lim ∈R(X/R, ∗) of ﬁnite pointed free C-spaces ←− R 5.7 Constant and Pseudoconstant Sheaves 175

(X/R, ∗), where R is a certain set of open equivalence relation on X (cf. RZ, Lemma 5.6.4(c)). Hence (see Corollary 5.7.5)

= G lim←− GR, R∈R ! where GR = ( K(X/R,∗R)(A)) C.LetR ∈ R. Since (X/R, ∗) is finite, the distinguished point ∗ is isolated, and so Corollary 5.7.8 applies; therefore GR = Ay (A∗R × C), y∈YR ∼ ∼ where YR is a certain subset of X/R and Ay = A = A∗R. GR is a free pro-p product whose free factors are all copies of A except one, ∼ which is A∗ × C(= A × Cp). (b) Let H be a finite subgroup of G and let HR denote its image in GR. Then HR ∼ ∼ is conjugate to a subgroup of a group the form A(= Cp) or A∗R × C(= Cp × Cp) 2 (this is proved later in Corollary 7.1.3); therefore, |HR|≤p . Hence the same holds for H . ∼ (c) By (a) L = A∗,C = Cp × Cp; similarly the image LR of L in GR is LR = ∼ A∗R,C = Cp ×Cp.Now,letM be another subgroup of G isomorphic to Cp ×Cp. ∼ Since M is finite, the collection R ={R ∈ R | ρR(M) = M} is cofinal in R.Let g TR ={g ∈ GR | MR = L }. Again using Corollary 7.1.3, we deduce that TR = ∅, R for R ∈ R ; and by Lemma 5.7.9, TR is compact. Then the inverse system {TR |

R ∈ R } has a nonempty inverse limit (see Sect. 1.1). Let g ∈ lim ∈R T . Then ←− R R M = Lg. Write g = ck (c ∈ C,k ∈ K). Then Lg = (A∗C)ck = (A∗C)k = Lk. (d) This is proved by a similar inverse limit argument.

Lemma 5.7.11 Let p be a fixed prime number and let B be a finite p-group. Let (X, ∗) be a pointed profinite space on which B acts freely. Consider the pseudo- K = K ∗ constant sheaf (X,∗)(A) over the! profinite pointed space (X, ) with fiber a torsion-free pro-p group A, and let K = K bethefreepro-p product of this sheaf. Let G = K B = K B be the corresponding semidirect product. Then (a) every finite subgroup of G is conjugate to a subgroup of B; and (b) NG(B) = B, where NG(B) denotes the normalizer of B in G.

Proof As in the lemma above, express G an inverse limit = = × G lim←− GR Ay (A∗R B), ∈R R y∈YR 176 5 Free Products of Pro-C Groups where each R is an open equivalence relation on X such that (X/R, ∗R) is a finite pointed space on which B acts freely. Note that in this case A∗R = 1, the trivial group, so that each GR is a free pro-p product of finitely many factors isomorphic to A and one isomorphic to B. Since A is torsion-free, every finite subgroup of GR is conjugate to a subgroup of B; and so, by the argument used in part (c) of the above = lemma, the same is true in G, proving (a). Now, NGR (B) B, by Corollary 7.1.6(b). So, using again a limit argument, NG(B) = B, proving (b). Chapter 6 Graphs of Pro-C Groups

6.1 Graphs of Pro-C Groups and Specializations

We begin with the notion of a graph of pro-C groups over a proﬁnite graph Γ .This is a way of associating to each element m of Γ apro-C group G(m) so that when m is an edge, there are two continuous monomorphisms of groups G(m) → G(dim) (i = 0, 1); and this is done taking into account the topological structure of Γ .We make this precise in the following formal deﬁnition.

Deﬁnition 6.1.1 Let Γ be a connected proﬁnite graph with incidence maps

d0,d1 : Γ −→ V(Γ).

A (proﬁnite) graph of pro-C groups over Γ is a sheaf (G,π,Γ)of pro-C groups over Γ together with two morphisms

(∂i,di) : (G,π,Γ)−→ GV ,π,V(Γ) (i = 0, 1) of sheaves [here GV denotes the restriction subsheaf of G to the space V(Γ), that we term the ‘vertex subsheaf of G’], where the restriction of ∂i to GV is the identity = map idGV (i 0, 1); in addition, we assume that the restriction of ∂i to each ﬁber G(m) is an injection (m ∈ Γ), (i = 0, 1).

∂i G G(m) GV G(di (m))

π π di Γ • V(Γ) • m di(m)

© Springer International Publishing AG 2017 177 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_6 178 6 Graphs of Pro-C Groups

Remark in view of the deﬁnition, it sufﬁces to specify the values of ∂i on GE(Γ ), and when E(Γ ) is closed in Γ , it is enough to verify the continuity of the functions ∂i on GE(Γ ). The vertex groups of a graph of pro-C groups (G,π,Γ)are the groups G(v), with v ∈ V(Γ), and the edge groups are the groups G(e), with e ∈ E(Γ ). Let (G,π,Γ)be a graph of pro-C groups over Γ , and let

ζ : Γ˜ −→ Γ be a universal Galois C-covering of the proﬁnite graph Γ (see Sect. 3.3). Choose a continuous 0-section j of ζ , and denote by J = j(Γ) the corresponding 0- transversal (see Lemma 3.4.3). Associated with j there is a continuous function

: −→ C χ Γ π1 (Γ ) from Γ into the fundamental pro-C group of Γ deﬁned by

χ(m) jd1(m) = d1j(m)

(see Eq. (3.1) in Sect. 3.4). Given a pro-C group H , deﬁne a J -specialization of the graph of pro-C groups (G,π,Γ)in H to consist of a pair (β, β ), where

β : (G,π,Γ)−→ H is a morphism from the sheaf (G,π,Γ)to H , and where

: C −→ β π1 (Γ ) H is a continuous homomorphism satisfying the following conditions:

−1 β(x) = β∂0(x) = β χ(m) β∂1(x) β χ(m) (6.1) for all x ∈ G, where m = π(x).

Note that if the set of edges E(Γ ) is closed in Γ , then it suffices to define β on the subsheaf GV(Γ) of vertices, because then one can extend the definition of β to thewholeofG by means of the formula β = β∂0 on GE(Γ ); in that case the first equality in (6.1) would be superfluous. C C = ˜ = = If Γ is -simply connected, then π1 (Γ ) 1 and Γ Γ J . Then J and the map β play no role, and in this case we refer to a ‘specialization’ rather than a ‘J -specialization’: it is just a morphism β : G −→ H such that

β(x) = β∂0(x) = β∂1(x) for all x ∈ G. 6.1 Graphs of Pro-C Groups and Specializations 179

Example 6.1.2 (a) Assume that the graph of pro-C groups (G,π,Γ) has trivial edge groups, i.e., G(e) = 1, for every edge e of Γ . Then we may think of a J - specialization (β, β ) of (G,π,Γ) in a pro-C group H as simply a morphism β from the sheaf (G,π,Γ)to H , since conditions (6.1) are automatic in this case. (b) As mentioned above, the case when Γ is a C-simply connected profinite graph is rather special. For example, if Γ is a finite tree or, more generally, an inverse limit of finite trees. Observe, however, that this is not necessarily the case for all C-trees Γ (see Example 3.10.6). (c) Assume now that Γ contains a spanning C-simply connected profinite subgraph T ; for example, this is the case if Γ a finite graph (but see Example 3.4.1). Then ζ : Γ˜ −→ Γ admits a fundamental 0-transversal J , i.e., J is a closed subset of Γ˜ which contains a profinite subgraph T that is mapped isomorphically to T by ζ , and d0(J ) ⊆ T (see Theorem 3.7.4). If one considers the function χ defined by the corresponding section j : Γ −→ Γ˜ , then clearly

χ(m)= 1 if and only if m ∈ T.

Therefore, with this choice of J , we may think of a J -specialization of a graph of pro-C groups (G,π,Γ)inapro-C group H as a morphism

β : (G,π,Γ)−→ H, together with a continuous map Γ −→ H , denoted m → sm (m ∈ Γ), such that sm = 1ifm ∈ T , and

= = −1 β(x) β∂0(x) sm β∂1(x) sm , for all x ∈ G(m). Indeed, just put sm = β χ(m). This way of expressing the concept of specialization perhaps clariﬁes the similarity with the theory for abstract graphs of groups (see, for example, Serre 1980, Sect. I.5.1). (d) Let Γ be a ﬁnite connected graph and let T be a maximal subtree of Γ .This is a particular case of (c). So, once we have chosen a fundamental 0-transversal J of Γ˜ → Γ as above, we can think of a J -specialization of the graph of groups (G,π,Γ)inapro-C group H as a collection of continuous homomorphisms

βv : G(v) → H, (v ∈ V(Γ)) together with a set {sm | m ∈ Γ } of elements of H such that sm = 1, for m ∈ T , and

= −1 βd0(e)∂0(x) se βd1(e)∂1(x) se , for all x ∈ G(e). (e) Let X be a proﬁnite space and consider the proﬁnite graph Γ = V(Γ)∪. E(Γ ), where V(Γ)={v} consists of a single vertex v, and E(Γ ) = X; i.e., Γ is a bouquet of X loops. This is a special case of (c). A graph of pro-C groups over Γ can then be viewed as follows: a sheaf (G,π,X) over the space X together with two morphisms ∂0,∂1 : G −→ Gv into a pro-C group Gv such that the restrictions of ∂0 and 180 6 Graphs of Pro-C Groups

∂1 to each ﬁber G(x) (x ∈ X) are monomorphisms. In this case a specialization in apro-C group H can be reinterpreted as consisting of a continuous homomorphism

β : Gv −→ H together with a continuous map β : X −→ Gv such that if we put = ∈ = −1 ∈ G ∈ tx β (x) (x X), then β∂0(g) tx(β∂1(g))tx , for all g (x) and x X.

6.2 The Fundamental Group of a Graph of Pro-C Groups

In this section we define a group associated with a graph of pro-C groups (G,π,Γ), its fundamental group, and prove its existence. Our definition depends initially on a choice of a 0-section of ζ : Γ˜ −→ Γ , but we will eventually prove that, in fact, it is independent of such a choice. Choose a continuous 0-section j of the universal Galois C-covering ζ : Γ˜ −→ Γ of Γ , and denote by J = j(Γ) the corresponding 0-transversal. We define a fundamental pro-C group of the graph of groups (G,π,Γ) with respect to the 0- C C G transversal J to be a pro- group Π1 ( ,Γ)together with a J -specialization (ν, ν ) G C G of ( ,π,Γ)in Π1 ( ,Γ)satisfying the following universal property:

C G C G Π1 ( ,Γ) Π1 ( ,Γ) ν ν

G δ C δ π1 (Γ )

β β HH whenever H is a pro-C group and (β, β ) a J -specialization of (G,π,Γ)in H , there exists a unique continuous homomorphism : C G −→ δ Π1 ( ,Γ) H such that δν = β and δν = β . We refer to (ν, ν ) as a universal J -specialization of (G,π,Γ). Observe that to check the above universal property it sufﬁces to consider only ﬁnite groups H in the pseudovariety C, since every pro-C group is an inverse limit of groups in C. C G In Theorem 6.2.4 we show that the notation Π1 ( ,Γ)is unambiguous when the C C G class is chosen appropriately because then the group Π1 ( ,Γ) does not depend on our choice of J .

Proposition 6.2.1 (Existence of fundamental groups) Let (G,π,Γ) be a graph of pro-C groups over a connected proﬁnite graph Γ , and let J be a 0-transversal of ζ : Γ˜ −→ Γ . Then 6.2 The Fundamental Group of a Graph of Pro-C Groups 181

C C G G (a) there exists a fundamental pro- group Π1 ( ,Γ)of ( ,π,Γ)with a universal J -specialization (ν, ν ); C G (b) (uniqueness for a ﬁxed J ) Π1 ( ,Γ)is unique in the sense that if Π is another fundamental pro-C group of (G,π,Γ), with respect to the same 0-transversal J , and (μ, μ ) is a universal J -specialization of (G,π,Γ) in Π, then there exists : C G −→ = a unique continuous isomorphism ξ Π1 ( ,Γ) Π such that ξν μ and ξν = μ ; C G { G | ∈ = } (c) Π1 ( ,Γ) is topologically generated by ν( (v)) v V V(Γ) and C ν (π1 (Γ )).

Proof Part (b) is an immediate consequence of the deﬁnition. Consider the free pro-C product (see Sect. 5.1)

W = G(v) = GV . v∈V

We identify each vertex group G(v) with its isomorphic image in W under the canonical morphism

ω : GV −→ W (see Proposition 5.1.6(c)). Deﬁne

C G = C Π1 ( ,Γ) W π1 (Γ ) /N, where denotes the free pro-C product of pro-C groups, and where N is the topo- C logical closure of the normal subgroup of the group W π1 (Γ ) generated by the set −1 −1 ∂0(x) χπ(x) ∂1(x) χπ(x) x ∈ G . : G −→ C G Deﬁne ν Π1 ( ,Γ)to be the composition of the natural continuous maps

G →∂0 G →ω → C → C G = C V W W π1 (Γ ) Π1 ( ,Γ) W π1 (Γ ) /N,

: C −→ C G and let ν π1 (Γ ) Π1 ( ,Γ) be the composition of the natural continuous homomorphisms

C → C → C G = C π1 (Γ ) W π1 (Γ ) Π1 ( ,Γ) W π1 (Γ ) /N.

Then one checks easily that (ν, ν ) is a J -specialization of the graph of pro-C groups G C G ( ,π,Γ)in Π1 ( ,Γ). C C G It follows immediately that the pro- group Π1 ( ,Γ)thus constructed, together with (ν, ν ), satisﬁes the universal property of a fundamental pro-C group of the graph of pro-C groups (G,π,Γ). This proves (a). Part (c) is clear from the construction above. 182 6 Graphs of Pro-C Groups

C = C Observe that if π1 (Γ ) 1, i.e., if Γ is a -simply connected proﬁnite graph, then there is only one 0-transversal J of ζ : Γ˜ −→ Γ , namely J = Γ˜ = Γ .Inthis C G case the notation Π1 ( ,Γ) is unambiguous since there is only one choice for J . The following corollary is an easy consequence of Proposition 6.2.1(c).

Corollary 6.2.2 Let (G,π,Γ) be a graph of pro-C groups over a C-simply con- C G nected proﬁnite graph Γ . Then Π1 ( ,Γ) is generated as a topological group by the images ν(G(v)) (v ∈ V(Γ))of the vertex groups.

Example 6.2.3 (a) Assume that the graph of pro-C groups (G,π,Γ) satisﬁes G(e) = 1, for all edges e of Γ . As pointed out in Example 6.1.2(a), a pair (β, β ) is a J -specialization in a pro-C group H if β is a morphism from G to H and β is a continuous homomorphism C C from π1 (Γ ) to H ; furthermore, J plays no role. In this case the fundamental pro- group of this graph of pro-C groups is just the free pro-C product

C G = G C Π1 ( ,Γ) (v) π1 (Γ ). v∈V(Γ)

We may take the universal specialization to be (ν, ν ), where ν is the composition ∼ G −→ G −→= G → G C (v) (v) π1 (Γ ), v∈V(Γ) v∈V(Γ) ! ! G −→ G : G −→ [the homomorphism v∈V(Γ) (v) is induced by ∂0 ∂V , and in this case it is an isomorphism], while ν is the natural inclusion

C → G C π1 (Γ ) (v) π1 (Γ ). v∈V(Γ) ! = C (b) Let G X Gx be a free pro- product. Then G can be viewed as the fundamental group of a graph of groups. To see this, first construct a profinite graph T = T(X) as follows. Its space of vertices is V(T)= X ∪. {ω}, where {ω} is a single point space disjoint from X, with the disjoint topology. Its space of edges is E(T ) ={ω}×X ={(ω, x) | x ∈ X}, with the product topology; the topology of T = V(T)∪. E(T ) is the disjoint topology. Finally, the incidence maps of T , di : T → V(T), are defined by d0(ω, x) = ω, d1(ω, x) = x(x∈ X), and d0 and d1 are the identity when restricted to V(T). Observe that T is in fact a C-tree; one can see this by expressing X as an inverse limit of finite spaces Xi and noticing = C that T(Xi) is a finite tree, and therefore T(X) ←−lim T(Xi) is a -tree (see Propo- sition 2.4.3(d)). Define G to be the subset of T × G consisting of those elements (m, y) ∈ T × G such that y ∈ Gm, if m ∈ X; y = 1, if m ∈{ω}∪. E(T ). 6.2 The Fundamental Group of a Graph of Pro-C Groups 183

It follows from Lemma 5.2.1 that G is a proﬁnite space, and so (G,π,T) is a subsheaf over T of the constant sheaf T × G. In fact, (G,π,T) has the structure of a graph of pro-C groups with obvious morphisms ∂0 and ∂1: they are identity maps on vertex ﬁbers and trivial homomorphisms otherwise.

Gx1 Gx 1 2

1 1 Gx3 1 . 1 . . Gx

C = Since T is an inverse limit of ﬁnite trees, π1 (T ) 1; hence, as a particular instance of Example 6.2.3(a), we have

= = C G G Gx Π1 ( ,T). X

(c) Assume that (G,π,Γ) is a graph of pro-C groups over a ﬁnite graph Γ .Let T be a maximal subtree of Γ .Let{te | e ∈ E(Γ )}. We think of the subset {te | e ∈ − } C C E(Γ ) E(T ) as a basis for the free pro- group π1 (Γ ). Then = C G = G Π Π1 ( ,Γ) (v) FC /N, v∈V(Γ)

C { | ∈ } where FC is the free pro- group! with basis te e E(Γ ) and N is the smallest G closed normal subgroup of ( v∈V(Γ) (v)) FC containing the set ∈ ∪ −1 −1 ∈ G ∈ te e E(T ) ∂0(x) te∂1(x)te x (e), e E(Γ ) .

C G ∈ Indeed, te plays the role of χ(e)in the original deﬁnition of Π1 ( ,Γ). For each v : G → = C G V(Γ),letνv (v) Π Π1 ( ,Γ) be the natural continuous homomorphism x → xN (x ∈ G(v)). The corresponding universal property described above that characterizes C G ∈ C { | ∈ }⊆ Π1 ( ,Γ) can now be stated as follows: whenever H , se e E(Γ ) H , with se = 1, ∀e ∈ E(T ) and βv : G(v) → H , v ∈ V(Γ), are continuous homo- = −1 ∀ ∈ G ∈ morphisms such that βd0(e)∂0(x) se(βd1(e)∂1(x))se , x (e), e E(Γ ), then : C G → = there exists a unique continuous homomorphism δ Π1 ( ,Γ) H with δνv βv, ∀v ∈ V(Γ)and δ(tm) = sm, ∀m ∈ Γ . [Note that, abusing notation, te denotes both C G the original element and its image teN in Π1 ( ,Γ).] 184 6 Graphs of Pro-C Groups

(d) If Γ consists of one edge e and two vertices v,w, the graph of groups (G,π,Γ)can be represented as

G G(e) G (v)• •(w)

In this case, J = Γ . The fundamental pro-C group of this graph of pro-C groups is an amalgamated free pro-C product C G = G G Π1 ( ,Γ) (v) G(e) (w) of the vertex groups G(v), G(w) with amalgamated edge subgroup G(e) (see Sect. 1.6). (e) Let Γ consist of one edge e and one vertex v,

G(v) • G(e)

In this case

C G = G G Π1 ( ,Γ) HNN (v), ∂0 (e) ,f is a pro-C HNN-extension of G(v) with associated subgroups ∂0(G(e)) and ∂1(G(e)), and where f : ∂0(G(e)) −→ ∂1(G(e)) is the isomorphism ∂0(x) → ∂1(x) (x ∈ G(e)) (cf. RZ, Sect. 9.4, where the notation is slightly different). More generally, consider a bouquet of X loops Γ with vertex v as in Exam- ple 6.1.2(e) and a corresponding graph of groups (G,π,Γ). Deﬁne

fx : Ax = ∂0 G(x) −→ Bx = ∂1 G(x)

= −1 ∈ G by fx(g) ∂1(∂0 (g)) (g ∂0 (x)). We term the corresponding fundamental pro- C C G group Π1 ( ,Γ) of such a graph of groups a (generalized) HNN extension with base group Gv and associated pairs of isomorphic subgroups Ax and Bx (x ∈ X); we denote it by

C G = ∈ Π1 ( ,Γ) HNN Gv,Ax,fx(x X) , together with a universal specialization (ν, ν ), i.e.,

ν : Gv −→ HNN Gv,Ax,fx(x ∈ X) is a continuous homomorphism and ν : X −→ HNN(Gv,Ax,fx(x ∈ X)) is a con- −1 tinuous map such that a = ν (x)fx(a)ν (x) for all a ∈ Ax , x ∈ X. It satisﬁes the following universal property: whenever H is a pro-C group and (β, β ) is a specialization in H , then there exists a unique continuous homomorphism

δ : HNN Gv,Ax,fx(x ∈ X) −→ H such that δν = β and δν = β . 6.2 The Fundamental Group of a Graph of Pro-C Groups 185

Uniqueness of the Fundamental Group

From now on in this section we shall assume that C is extension-closed.

C C G Our next aim is to obtain a description of the fundamental pro- group Π1 ( ,Γ) that is independent of the choice of J . We do this by expressing this fundamental C group in terms of π1 (Γ ) and the fundamental group of a certain graph of groups over the universal Galois C-covering graph Γ˜ of Γ , which is known to be C-simply connected (see Theorem 3.7.1(a)), and hence, as we have pointed out above, does not require any choice when dealing with a transversal. Consider the pull-back

π˜ G˜ Γ˜

ζ˜ ζ

G Γ π of the maps π : G −→ Γ and ζ : Γ˜ −→ Γ . In other words, G˜ = (x, m)˜ ∈ G × Γ˜ π(x) = ζ(m),˜ x ∈ G, m˜ ∈ Γ˜ ⊆ G × Γ.˜

Define maps π˜ : G˜ −→ Γ˜ and ζ˜ : G˜ −→ G to be the restrictions to G˜ of the canonical projections from G × Γ˜ to Γ˜ and G, respectively. For a fixed m˜ ∈ Γ˜ , define

G˜(m)˜ =˜π −1(m)˜ = G ζ(m)˜ ×{˜m}, which is a group with respect to the operation

(x, m)˜ x , m˜ = xx , m˜ x,x ∈ G ζ(m)˜ .

Clearly

G˜(m)˜ =∼ G ζ(m)˜ .

Hence (G˜, Γ)˜ is a graph of pro-C groups over the proﬁnite graph Γ˜ , with morphisms

˜ ˜ ˜ ˜ ˜ (∂i,di) : (G, π,˜ Γ)−→ GV , π,V(˜ Γ) (i = 0, 1), where the maps ˜ ˜ ˜ ∂i : G −→ GV (i = 0, 1) 186 6 Graphs of Pro-C Groups are given by

˜ ˜ ∂i(x, m)˜ = ∂i(x), di(m)˜ , for (x, m)˜ ∈ G (i = 0, 1) ˜ (observe that π˜ ∂i = diπ˜ for i = 0, 1). C G˜ Note that there is a natural continuous action of π1 (Γ ) on deﬁned by ˜ = ˜ ∈ C ˜ ∈ G˜ h(x, m) (x, hm), for h π1 (Γ ), (x, m) . ∈ C Associated with each h π1 (Γ ), we have a continuous function

h¯ : G˜ −→ G˜ given by

h(x,¯ m)˜ = h(x, m)˜ = (x, hm)˜ (x, m)˜ ∈ G˜ . Observe that Γ˜ is connected by definition, and it is also C-simply connected (see C G˜ ˜ Theorem 3.7.1(a)). Therefore, as pointed out above, Π1 ( , Γ)is well-defined for there is no need to make a choice of a 0-transversal J . Denote by ˜ : G˜ −→ C G˜ ˜ ν Π1 ( , Γ) the corresponding universal specialization, so that ˜ ˜ ˜ ν(x,˜ m)˜ =˜ν∂0(x, m)˜ =˜ν∂1(x, m),˜ for all (x, m)˜ ∈ G. C We shall define next a continuous action of the group π1 (Γ ) on the group C G˜ ˜ Π1 ( , Γ). In order to do this, we note first that ˜ ¯ =˜¯ ˜ =˜¯ ˜ ∈ C ; νh νh∂0 νh∂1, for all h π1 (Γ ) i.e., ˜ ¯ : G˜ −→ C G˜ ˜ νh Π1 ( , Γ) is a specialization. Therefore, by the universal property of the fundamental group, there exists a unique continuous homomorphism : C G˜ ˜ −→ C G˜ ˜ λh Π1 ( , Γ) Π1 ( , Γ) such that the following diagram

ν˜ G˜ C G˜ ˜ Π1 ( , Γ)

λh ν˜h¯ C G˜ ˜ Π1 ( , Γ) 6.2 The Fundamental Group of a Graph of Pro-C Groups 187 commutes, i.e., ˜ =˜¯ ∈ C λhν νh, for all h π1 (Γ ). (6.2) ∈ C Let h1,h2 π1 (Γ ).Using(6.2) we get ˜ = ˜ ¯ =˜¯ ¯ =˜ = ˜ λh2 λh1 ν λh2 νh1 νh2h1 νh2h1 λh2h1 ν. = By uniqueness we deduce that λh2 λh1 λh2h1 , i.e., the map

: C −→ C G˜ ˜ λ π1 (Γ ) End Π1 ( , Γ) deﬁned by h → λh is multiplicative. Furthermore, λ1 = id C G˜ ˜ . Therefore, Π1 ( ,Γ) = −1 ∈ C λh−1 λh , for all h π1 (Γ ). ∈ C G˜ ˜ ∈ C Hence λh Aut(Π1 ( , Γ)), for all h π1 (Γ ).Sothemap

: C −→ C G˜ ˜ λ π1 (Γ ) Aut Π1 ( , Γ) → C G˜ ˜ deﬁned by h λh is a homomorphism into Aut(Π1 ( , Γ)). We deﬁne an action C × C G˜ ˜ −→ C G˜ ˜ π1 (Γ ) Π1 ( , Γ) Π1 ( , Γ) C C G˜ ˜ of the group π1 (Γ ) on the group Π1 ( , Γ)as follows:

→ h = ∈ C ∈ C G˜ ˜ (h, g) g λh(g) h π1 (Γ ), g Π1 ( , Γ) . We claim that this action is continuous or, equivalently, that the homomorphism λ C G˜ ˜ is continuous, where Aut(Π1 ( , Γ))is endowed with the compact-open topology C G˜ ˜ To see this, let U be an open normal subgroup of Π1 ( , Γ), and let : C G˜ ˜ −→ = C G˜ ˜ α Π1 ( , Γ) ΠU Π1 ( , Γ)/U denote the corresponding canonical epimorphism. Since ΠU is ﬁnite, Ap = −1 ˜ (αν)˜ (p) is a clopen subset of G for every p ∈ ΠU .

α C G˜ ˜ = C G˜ ˜ Π1 ( , Γ) ΠU Π1 ( , Γ)/U ν˜ ν˜

n¯ G˜ G˜

C G˜ Since Ap is compact and since π1 (Γ ) acts continuously on , there exists an open C normal subgroup Np of π1 (Γ ) such that

NpAp ⊆ Ap. 188 6 Graphs of Pro-C Groups

Put N = Np. p∈ΠU

So N is an open normal subgroup of ΠU and

NAp ⊆ Ap, for all p ∈ ΠU . From this and (6.2) we deduce that

αν˜ = αν˜n¯ = αλnν,˜ for all n ∈ N. C G˜ ˜ From the uniqueness condition of the universal property deﬁning Π1 ( , Γ)we have

α = αλn, for all n ∈ N. This means that n = ∈ ∈ ∈ ∈ C G˜ ˜ (ux) λn(ux) Ux, for all u U,n N,x Π1 ( , Γ), (6.3) ∈ C G˜ ˜ i.e., the coset Ux is invariant under the action of N, for each x Π1 ( , Γ).In particular, the subgroup U is N-invariant, i.e.,

n λn(U) = U = U, for all n ∈ N. C Note that since N has finite index in π1 (Γ ), there exist only finitely many (open) C G˜ ˜ h ∈ C subgroups of Π1 ( , Γ)of the form U(h π1 (Γ )). Therefore W = hU ∈ C h π1 (Γ ) C G˜ ˜ C is an open normal subgroup of Π1 ( , Γ) contained in U, and it is π1 (Γ )- ∈ C invariant. Hence, using this and (6.3), for any given fixed elements h π1 (Γ ) ∈ C G˜ ˜ C and g Π1 ( , Γ), the open neighbourhoods Nh of h in π1 (Γ ) and Wg of g in C G˜ ˜ Π1 ( , Γ)satisfy

Nh(Wg) = W Nhg ⊆ U hg . C C G˜ ˜ Thus the action of π1 (Γ ) on Π1 ( , Γ)is continuous, proving the claim. Using this action we deﬁne a semidirect product = C G˜ ˜ C G Π1 ( , Γ) π1 (Γ ). This is a pro-C group since C is extension-closed. Note that the deﬁnition of G does not require any choice of transversal of ζ : Γ˜ → Γ .InG the formula (6.2) becomes − hν(x,˜ m)˜ = hν(x,˜ m)h˜ 1 =˜ν(x,hm),˜ (6.4) ∈ C ∈ G ˜ ∈ ˜ = ˜ where h π1 (Γ ), x , m Γ,π(x) ζ(m). 6.2 The Fundamental Group of a Graph of Pro-C Groups 189

Fix a continuous 0-section j of ζ : Γ˜ −→ Γ . Deﬁne a map

: G −→ = C G˜ ˜ C σj G Π1 ( , Γ) π1 (Γ ) by

=˜ ∈ C G˜ ˜ ≤ ∈ G ; σj (x) ν x,jπ(x) Π1 ( , Γ) G(x ) (6.5) and let : C −→ σj π1 (Γ ) G be the inclusion homomorphism. In the following theorem we continue with this set-up and notation.

Theorem 6.2.4 (Uniqueness of fundamental group of a graph of groups) Let (G,π,Γ)be a graph of pro-C groups over a connected proﬁnite graph Γ . (a) For a continuous 0-section j of ζ : Γ˜ −→ Γ with corresponding 0-transversal = J j(Γ), the pair (σj ,σj ) is a J -specialization of the graph of groups (G,π,Γ)in G. C (b) The group G together with the J -specialization (σj ,σj ) is a fundamental pro- group of the graph of pro-C groups (G,π,Γ)with respect to the 0-transversal J . ˜ (c) If j1 is another continuous 0-section of ζ : Γ −→ Γ , there exists a function f : Γ → G such that

= −1 ∈ ∈ G σj1 (x) f(m) σj (x)f (m) m Γ,x (m) .

Proof (a) Indeed, for y ∈ G we have (using the identity (6.4) in the seventh equality)

σj ∂0(y) =˜ν ∂0(y), jπ∂0(y) =˜ν ∂0(y), d0jπ(y)

˜ ˜ =˜ν∂0 y,jπ(y) =˜ν∂1 y,jπ(y)

=˜ν ∂1(y), d1jπ(y) =˜ν ∂1(y), χπ(y) jd1π(y)

−1 = χπ(y) ν˜ ∂1(y), jd1π(y) χπ(y)

−1 = χπ(y) ν˜ ∂1(y), jπ∂1(y) χπ(y)

= −1 σj χπ(y) σj ∂1(y) σj χπ(y) and

˜ σj (y) =˜ν y,jπ(y) =˜ν∂0 y,jπ(y) =˜ν ∂0(y), d0jπ(y)

=˜ν ∂0(y), jπ∂0(y) = σj ∂0(y).

C G C (b) Let Π1 ( ,Γ) denote the fundamental pro- group of the graph of pro- C groups (G,π,Γ) with universal J -specialization (ν, ν ) constructed in Proposi- 190 6 Graphs of Pro-C Groups

tion 6.2.1.TheJ -specialization (σj ,σj ) determines a unique continuous homomorphism : C G −→ αj Π1 ( ,Γ) G, = = such that αj ν σj and αj ν σj . It sufﬁces then to show that αj is an isomorphism, and to do this we will show that it has an inverse homomorphism : → C G αj G Π1 ( ,Γ).

Before deﬁning αj , we shall obtain an auxiliary continuous homomorphism

: C G˜ ˜ −→ C G α Π1 ( , Γ) Π1 ( ,Γ). To do this consider the map : G˜ −→ C G τ Π1 ( ,Γ) deﬁned by

−1 ˜ τ(x,m)˜ = ν κj (m)˜ ν(x) ν κj (m)˜ x ∈ G, m˜ ∈ Γ,π(x)= ζ(m)˜ , where : ˜ −→ C κj Γ π1 (Γ ) is the function deﬁned by the equalities (see Eq. (3.1) in Sect. 3.4)

˜ κj (m)˜ jζ(m)˜ =˜m(m˜ ∈ Γ).

G˜ ˜ C G We claim that τ is a specialization of the graph of groups ( , Γ) in Π1 ( ,Γ). Using Lemma 3.4.4(c), one easily sees that ˜ ˜ τ(x,m)˜ = τ∂0(x, m),˜ for every (x, m)˜ ∈ G.

To verify the claim we still need to prove that ˜ ˜ τ∂0 = τ∂1.

To see this, let (x, m)˜ ∈ G˜; then, using Lemma 3.4.4(c) and (e),

˜ −1 τ∂0(x, m)˜ = τ ∂0(x), d0(m)˜ = ν κj d0(m)˜ ν∂0(x) ν κj d0(m)˜

−1 = ν κj (m)˜ ν∂0(x) ν κj (m)˜

−1 −1 = ν κj (m)˜ ν χζ(m)˜ ν∂1(x) ν χζ(m)˜ ν κj (m)˜

−1 = ν κj d1(m)˜ ν∂1(x) ν κj d1(m)˜

˜ = τ ∂1(x), d1(m)˜ = τ∂1(x, m).˜

This completes the veriﬁcation of the claim. 6.2 The Fundamental Group of a Graph of Pro-C Groups 191

Therefore τ determines a unique continuous homomorphism : C G˜ ˜ −→ C G α Π1 ( , Γ) Π1 ( ,Γ) such that α ν˜ = τ . Extend α toamap : = C G˜ ˜ C −→ C G αj G Π1 ( , Γ) π1 (Γ ) Π1 ( ,Γ) given by

= ∈ C G˜ ˜ ∈ C αj (gh) α (g)ν (h) g Π1 ( , Γ),h π1 (Γ ) .

Clearly αj is continuous. We assert that αj is a homomorphism. To prove this we ∈ C ∈ C G˜ ˜ need to verify that if h π1 (Γ ) and g Π1 ( , Γ), then

α hg = ν (h)α (g)ν (h)−1.

In view of Corollary 6.2.2, to check this we may assume that g =˜ν(x,m)˜ ,forsome (x, m)˜ ∈ G˜. We know (see Eq. (6.2)) that

hν(x,˜ m)˜ =˜ν(x,hm).˜

So, using Lemma 3.4.4(b), we have

h −1 α g = α ν(x,h˜ m)˜ = τ(x,hm)˜ = ν κj (hm)˜ ν(x) ν κj (hm)˜

−1 = ν hκj (m)˜ ν(x)ν hκj (m)˜

−1 −1 −1 = ν (h) ν κj (m)˜ ν(x) ν κj (m)˜ ν (h) = ν (h)τ(x, m)ν˜ (h)

= ν (h) α ν(x,˜ m)˜ ν (h)−1 = ν (h)α (g)ν (h)−1, as needed. Hence the assertion is proved. ∈ G Our ﬁnal step is to show that αj is an isomorphism (inverse to αj ). Let x . Then

= = = ˜ = αj αj ν(x) αj σj (x) α σj (x) α ν x,jπ(x) τ x,jπ(x)

−1 = ν κj jπ(x) ν(x) ν κj jπ(x) = ν(x), = = since κj (jπ(x)) 1 (see Lemma 3.4.4(d)). Therefore, αj αj ν ν, and so,

α αj = id C G . j Π1 ( ,Γ )

It follows that αj is an injection. ≥ C To prove that αj is also a surjection, we observe ﬁrst that Im(αj ) π1 (Γ ) since = by construction αj ν σj and : −→ = C G˜ ˜ C σj π1(Γ ) G Π1 ( , Γ) π1 (Γ ) 192 6 Graphs of Pro-C Groups is the inclusion. Therefore it sufﬁces to show that Im(ν)˜ ⊆ Im(αj ), since C G˜ ˜ = ˜ Π1 ( , Γ) Im(ν) ,

˜ according to Corollary 6.2.2. To ﬁnish the proof, we verify this. For (x, m)˜ ∈ GV , we use (6.4) and the fact that π(x) = ζ(m)˜ to deduce that

−1 ν(x,˜ m)˜ =˜ν x,κj (m)˜ jζ(m)˜ = κj (m)˜ ν˜ x,jζ(m)˜ κj (m)˜ −1 = κj (m)σ˜ j (x)κj (m)˜

−1 = αj ν κj (m)˜ ν(x) ν κj (m)˜ . (6.6)

Therefore we have shown that αj is a surjection, and so a continuous isomorphism

(whose inverse is αj ). −1 (c) From (6.6) we have that ν(x,˜ m)˜ = κj (m)σ˜ j (x)κj (m)˜ , where π(x) = ζ(m)˜ . ˜ ˜ = ˜ ˜ −1 Similarly, for the 0-section j1 we get ν(x,m) κj1 (m)σj1 (x)κj1 (m) . Hence = ˜ −1 ˜ ˜ −1 ˜ −1 = ˜ = σj1 (x) κj1 (m) κj (m)σj (x)(κj1 (m) κj (m)) .Letm π(x), and choose m j(m). By Lemma 3.4.4(d), κj (j (m)) = 1. Thus,

= −1 ∈ ∈ G σj1 (x) f(m) σj (x)f (m), m Γ,x (m) ,

= where f(x) κj1 (j (m)).

C G The theorem above shows that the group Π1 ( ,Γ)of Proposition 6.2.1 is well- deﬁned: it corresponds to the pro-C group G in the above theorem, whose deﬁnition is not dependent on the choice of a 0-section j of ζ : Γ˜ → Γ . We also get the following consequence.

C G C G = Corollary 6.2.5 Π1 ( ,Γ)contains a subgroup isomorphic to π1 (Γ ). If (m) 1 ∈ C G = C for every m Γ , then Π1 ( ,Γ) π1 (Γ ).

Remark 6.2.6 Unlike the situation for graphs of abstract groups, the canonical ho- : G → C G momorphisms νv (v) Π1 ( ,Γ) are not injective in general (see Sect. 6.4 below). However, one can show that in some important cases they are, e.g., when dealing with free products (see Proposition 5.1.6), or when dealing with graphs of finite groups over finite graphs (see Proposition 6.5.6 below). The injectivity of the maps νv is of crucial importance, especially when trying to relate properties of abstract graphs of groups with properties of profinite graphs of profinite groups, for example in Sect. 11.3 below or in the arguments in Chap. 15, where injectivity in the case of free product with amalgamation becomes one of the central prerequisites in many results. 6.3 The Standard Graph of a Graph of Pro-C Groups 193

6.3 The Standard Graph of a Graph of Pro-C Groups

Let (G,π,Γ) be a graph of pro-C groups over a connected profinite graph Γ . Let j : Γ −→ Γ˜ be a continuous 0-section of the universal Galois C-covering ζ : Γ˜ −→ Γ of Γ , and let J = j(Γ) be the corresponding 0-transversal (see Sect. 3.4). Let (γ, γ ) be a J -specialization of (G,π,Γ)inapro-C group H . In the first place we shall define a certain profinite graph C S (G,Γ,H) which is canonically associated to the graph of groups (G,π,Γ)and H . For m ∈ Γ , define H(m)= γ(G(m)). Then the collection H(m) m ∈ Γ of closed subgroups of H is continuously indexed by Γ . Indeed, if U is an open subset of H , then γ −1(H − U) is a compact subset of G; since Γ − m ∈ Γ H(m)⊆ U = π γ −1(H − U) , it follows that Γ(U)= m ∈ Γ H(m)⊆ U is open in Γ . As a topological space, SC(G,Γ,H)is defined to be the quotient space of Γ ×H modulo the equivalence relation ∼ given by

− (m, h) ∼ m ,h if m = m ,h 1h ∈ H(m) m, m ∈ Γ, h,h ∈ H . (6.7)

So, as a set, SC(G,Γ,H)is the disjoint union C S (G,Γ,H)= . H/H(m). m∈Γ

By Proposition 5.2.3, SC(G,Γ,H)is a profinite space. Denote by C α : Γ × H −→ S (G,Γ,H) the quotient map. The projection p : Γ × H −→ Γ induces a continuous epimorphism, which we again denote by p, C p : S (G,Γ,H)−→ Γ, such that p−1(m) = H/H(m) (cf. Bourbaki 1989, I, 3, 4, Proposition 6). To make SC(G,Γ,H) into a profinite graph we define the subspace of vertices of SC(G,Γ,H)by

C − V S (G,Γ,H) = p 1 V(Γ) . 194 6 Graphs of Pro-C Groups

The incidence maps C C di : S (G,Γ,H)−→ V S (G,Γ,H) (i = 0, 1) are deﬁned as follows:

d0 hH (m) = hH d0(m)

(6.8) d1 hH (m) = h γ χ(m) H d1(m) ,(h∈ H, m ∈ Γ),

: −→ C where χ Γ π1 (Γ ) is the continuous map considered in Eq. (3.1) in Sect. 3.4. We must check that these maps are well-deﬁned and continuous. The map d0 is well-deﬁned since H(m) ≤ H(d0(m)), for all m ∈ Γ .Lety ∈ H(m) and h ∈ H ; then

d1 hyH (m) = hy γ χ(m) H d1(m) .

To see that d1 is well-deﬁned we need to check that

h γ χ(m) H d1(m) = hy γ χ(m) H d1(m) , or, equivalently, that

−1 −1 y ∈ γ χ(m) H d1(m) γ χ(m) = γ χ(m) γ G d1(m) γ χ(m) .

But, according to (6.1), this is true since y = γ(x),forsomex ∈ G(m), and ∂1(x) ∈ G(d1(m)). To verify that the maps d0 and d1 are continuous, observe ﬁrst that since V(Γ)× H is closed in Γ × H , the restriction αV of α to V(Γ)× H is the quotient map αV : V(Γ)× H −→ V(Γ)with respect to the (restriction of the) equivalence relation ∼. Consider the commutative diagrams

¯ di Γ × H V(Γ)× H

α αV

SC(G,Γ,H) V(SC(G,Γ,H)) di

¯ ¯ (i = 0, 1), where d0(m, h) = (d0(m), h) and d1(m, h) = (d1(m), (γ χ(m))h) (m ∈ Γ, h ∈ H). Since the topologies of SC(G,Γ,H) and V(SC(G,Γ,H)) are quotient topologies, to check the continuity of d0 and d1 it suffices to prove that the ¯ ¯ ¯ ¯ maps d0 and d1 are continuous. For d0 this is obvious, and for d1 it follows from the continuity of γ , χ and of the multiplication in H . This completes the definition of SC(G,Γ,H). The profinite graph C S (G,Γ,H) 6.3 The Standard Graph of a Graph of Pro-C Groups 195 is called the profinite C-standard graph of the graph of pro-C groups (G,π,Γ)with respect to H or the C-universal covering of the graph of pro-C groups (G,π,Γ) with respect to H . There is a natural continuous action of H on the graph SC(G,Γ,H)given by

h h H(m) = hh H(m) h, h ∈ H, m ∈ Γ .

C C G Assume now that is extension-closed so that Π1 ( ,Γ)is well-deﬁned. If = = C G H Π Π1 ( ,Γ) and (γ, γ ) = (ν, ν ) is the universal J -specialization of the graph of pro-C groups G = C G = C G ( ,π,Γ) in Π Π1 ( ,Γ), we use instead the notation S S ( ,Γ) rather than SC(G,Γ,Π), and we refer to C S = S (G,Γ)= . Π/Π(m) Π(m)= ν G(m) ,m∈ Γ (6.9) m∈Γ as the C-standard graph (or C-universal graph) of the graph of pro-C groups (G,π,Γ).1 We sometimes write S(G,Γ) instead of SC(G,Γ) if there is no danger of confusion.

Example 6.3.1 Assume Γ is ﬁnite and let (G,Γ) be a graph of pro-C groups over Γ . Using the notation of Example 6.2.3(c), the C-standard graph S = SC(G,Γ) has vertices and edges V(S)= Π/Π(v) and E(S) = Π/Π(e) v∈V(Γ) e∈E(Γ ) and incidence maps

d0 gΠ(e) = gΠ d0(e) ,d1 gΠ(e) = gteΠ d1(e) g ∈ Π,e ∈ E(Γ ) .

C = C G = Lemma 6.3.2 Assume that is extension-closed. Let Π Π1 ( ,Γ) and S SC(G,Γ)be as above. (a) The quotient space Π\S is Γ . Furthermore, the Π-stabilizer of a point gΠ(m) of S is gΠ(m)g−1. (b) The map σ : Γ −→ S given by

σ(m)= 1Π(m), (m∈ Γ)

is a continuous section of p : S −→ Γ , as a map of topological spaces.

1In Serre (1980), Sect. I.5.3, Serre refers to the corresponding concept in the context of abstract groups, as a ‘universal covering’ graph relative to a graph of groups. In the present book, we occasionally use the expression ‘standard graph’ of a graph of groups, even when referring to abstract groups; the meaning should be clear by the context. 196 6 Graphs of Pro-C Groups

(c) Assume that Γ is C-simply connected. Then the section σ defined in (b) is a monomorphism of profinite graphs; in particular, in this case Γ is embedded as a profinite subgraph of S. (d) Assume that Γ admits a spanning profinite subgraph T which is C-simply connected. Then the map σ : Γ −→ S given by

σ(m)= 1Π(m), (m∈ Γ)

is a fundamental 0-section of p : S −→ Γ lifting T .

Proof Part (a) is clear. Note that the map σ1 : Γ −→ Γ × Π given by σ1(m) = (m, 1)(m∈ Γ)is a continuous section of the natural projection Γ × Π −→ Γ .Let σ be the composition of σ1 and the quotient map

α : Γ × Π −→ S, i.e., σ(m) = 1Π(m) (m ∈ Γ). Then σ is a continuous section of p, as maps of topological spaces, proving (b). Part (c) is a consequence of (d) when T = Γ . We next prove (d). To show that σ is a fundamental 0-section of p lifting T , observe first that d0(1Π(m)) = 1Π(d0(m)) ∈ σ(T) for every m ∈ Γ . Hence it suffices to show that the injection σ|T : T −→ S is a morphism of graphs. This follows from the fact that the function χ used in formula (6.8) can be chosen so that χ(m) = 1ifm ∈ T (see Theo- rem 3.7.4(a) and the definition of χ).

The graph S = SC(G,Γ) deﬁned above is given in terms of the universal J - specialization (ν, ν ). The next result shows that the C-standard graph of the graph of pro-C groups (G,π,Γ) is independent of the choice of J , up to isomorphism, when C is extension-closed. If Γ is C-simply connected, then J = Γ˜ = Γ , and the reference to J is unnecessary; in this case SC(G,Γ)is clearly well-deﬁned.

Theorem 6.3.3 (Uniqueness of the standard graph) Assume that C is extension- closed. Let (G,π,Γ)be a graph of pro-C groups over a connected profinite graph Γ . Choose a continuous 0-section j : Γ −→ Γ˜ of the universal Galois C-covering ζ : Γ˜ −→ Γ . Then there exists an isomorphism of profinite graphs C C Φ : S˜ = S (G˜, Γ)˜ −→ S = S (G,Γ), where (G˜, Γ)˜ is the graph of pro-C groups constructed in Sect. 6.2. In particular, the definition of a C-standard graph of a graph of pro-C groups is independent, up to isomorphism, of the choice of j.

Proof As in Sect. 6.2,letν˜ be a universal C-specialization of the graph of groups G˜ ˜ C ˜ ˜ = C G˜ ˜ ( , Γ) (over the -simply connected graph Γ )inΠ Π1 ( , Γ). According to = C G = ˜ C Theorem 6.2.4, Π Π1 ( ,Γ) can be identiﬁed with the group G Π π1 (Γ ) C together with the universal -specialization (σj ,σj ) deﬁned by the equalities (6.5). 6.3 The Standard Graph of a Graph of Pro-C Groups 197

Deﬁne maps of topological spaces (for the deﬁnition of κj ,seeEq.(3.1)in Sect. 3.4)

Φ1 Γ˜ × Π˜ Γ × G

Φ1 as follows:

˜ ˜ Φ1(m,˜ g)˜ = ζ(m),˜ gκ˜ j (m)˜ (m˜ ∈ Γ,g˜ ∈ Π) ∈ ∈ =˜ ˜ ∈ ˜ ∈ C and, if m Γ , g G, with g gh, where g Π and h π1 (Γ ),

= ˜ Φ1(m, g) hj(m), g . These maps are clearly continuous and they are easily seen to be inverse to each other. We claim that Φ1 (respectively, Φ1) is compatible with the equivalence relation ∼ on Γ˜ × G˜ (respectively, Γ × G) deﬁned in (6.7). Let m˜ ∈ Γ˜ , g˜ ∈ Π˜ and t =˜ν(x,m)˜ , where x ∈ G(ζ(m))˜ (i.e., ζ(m)˜ = π(x)). We need to show that Φ1(m,˜ gt)˜ ∼ Φ1(m,˜ g)˜ . Before we do this observe that

σj (x) =˜ν x,jπ(x) =˜ν x,jζ(m)˜

−1 −1 =˜ν x,κj (m)˜ m˜ = κj (m)˜ ν(x,˜ m)κ˜ j (m).˜

Now, using this, we obtain

Φ1(m,˜ gt)˜ = Φ1 m,˜ g˜ν(x,˜ m)˜ = ζ(m),˜ g˜ν(x,˜ m)κ˜ j (m)˜

−1 = ζ(m),˜ gκ˜ j (m)˜ κj (m)˜ ν(x,˜ m)κ˜ j (m)˜

= ζ(m),˜ gκ˜ j (m)σ˜ j (x) ∼ ζ(m),˜ gκ˜ j (m)˜ = Φ1(m,˜ g),˜ as required. One checks similarly that Φ is compatible with ∼. 1 Hence Φ1 and Φ1 induce maps of topological spaces

C Φ C S˜ = S (G˜, Γ)˜ = (Γ˜ × Π/˜ ∼) S = S (G,Γ)= (Γ × G/ ∼) Φ that are inverse to each other. Therefore Φ is a homeomorphism of topological spaces. Next we show that Φ is also a morphism of proﬁnite graphs, and so an isomorphism. To do this note that according to (6.9), an element of S˜ can be written as g˜Π(˜ m)˜ (g˜ ∈ Π,˜ m˜ ∈ Π˜ ). With this notation we have

˜ Φ g˜Π(m)˜ =˜gκj (m)Π˜ ζ(m)˜ .

Then, using Lemma 3.4.4(c),

˜ ˜ Φd0 g˜Π(m)˜ = Φ g˜Π d0(m)˜ =˜gκj d0(m)˜ Π ζd0(m)˜

˜ =˜gκj (m)Π˜ d0ζ(m)˜ = d0Φ g˜Π(m)˜ . 198 6 Graphs of Pro-C Groups

Also, using Lemma 3.4.4(e) in the third equality, we have

˜ ˜ Φd1 g˜Π(m)˜ = Φ g˜Π d1(m)˜ =˜gκj d1(m)Π˜ ζd1(m)˜

=˜gκj (m)χζ(˜ m)Π˜ d1ζ(m)˜

=˜ ˜ ˜ ˜ = ˜ ˜ ˜ gκj (m) σj χζ(m) Π d1ζ(m) d1Φ gΠ(m) .

Proposition 6.3.4 Assume that C is extension-closed. Let (G,π,Γ) be a graph of pro-C groups over a connected proﬁnite graph Γ , and let S = SC(G,Γ) be the corresponding C-standard graph. Then S is connected as a proﬁnite graph.

Proof By Theorem 6.3.3 and Theorem 3.7.1(a) we may assume that Γ is C-simply connected. Let ϕ : S −→ be an epimorphism of profinite graphs where is a finite graph. We need to show that is connected as an abstract graph. We write S as in (6.9). Let δ ∈ . Since ϕ−1(δ) is open in S, for each gΠ(m) ∈ −1 ⊆ ∈ = C G ∈ ϕ (δ) S(g Π Π1 ( ,Γ), m Γ), there exists an open normal subgroup δ δ Ug,m of Π and an open neighbourhood Vg,m of m in Γ such that the open neighbourhood ∈ δ ∈ δ guΠ(n) u Ug,m,n Vg,m of gΠ(m) in S is contained in ϕ−1(δ). Since ϕ−1(δ) is compact, there is a finite −1 subset {g1Π(m1),...,gr(δ)Π(mr(δ))} of ϕ (δ) such that the collection of open subsets g uΠ(n) u ∈ U δ ,n∈ V δ i = ,...,r(δ) i gi ,mi gi ,mi 1 of S covers ϕ−1(δ). Put r(δ) U = U δ . gi ,mi δ∈ i=1 Then U is an open normal subgroup of Π and

ϕ gΠ(m) = ϕ guΠ(m) , for every u ∈ U, m ∈ Γ.

Let v = ϕ(gΠ(m)) and w = ϕ(hΠ(n)) (g, h ∈ Π, m,n ∈ Γ)be arbitrary vertices in . We shall show that v and w are contained in some connected subgraph of . This will prove that is connected. To do this, observe ﬁrst that the neighbourhood g−1hU of g−1h in Π contains an element, say x, of the abstract subgroup generated by the subgroups Π(v) (v ∈ V(Γ)) (see Corollary 6.2.2). Then x = x1 ···xt ,forsomexi ∈ Π(vi), where v1,...,vt ∈ V(Γ). Put

g0 = g, gi = gi−1xi (i = 1,...,t), so that hU = gxU = gt U, and hence w = ϕ(hΠ(n)) = ϕ(gt Π(n)). 6.3 The Standard Graph of a Graph of Pro-C Groups 199

Let σ : Γ −→ S be the embedding of Γ in S described in Lemma 6.3.2. Consider the subgraph t Σ = giσ(Γ) i=0 of S. Observe that

gi−1xiΠ(mi) = giΠ(mi) ∈ gi−1σ(Γ)∩ giσ(Γ), (i = 1,...,t). So Σ is a proﬁnite connected graph (see Lemma 2.1.7(b)). Therefore ϕ(Σ) is a connected subgraph of . Since v,w ∈ ϕ(Σ), there is a path joining v and w in ; i.e., is connected.

Theorem 6.3.5 Assume that C is extension-closed. Let (G,π,Γ) be a graph of pro-C groups over a connected proﬁnite graph Γ and let S = SC(G,Γ) be the corresponding C-standard graph. Then S is a C-simply connected proﬁnite graph.

Proof By Theorem 6.3.3 and Theorem 3.7.1(a) we may assume that Γ is C-simply connected. Denote by : G −→ = C G ν Π Π1 ( ,Γ) a universal specialization. By Theorem 3.7.1 it suffices to prove that the identity map idS : S → S is a universal Galois C-covering. To prove this, consider a finite graph and a morphism ϕ : S = . Π/Π(m) −→ m∈Γ of profinite graphs, and let ζ : Σ −→ be a finite Galois C-covering of .Fix z ∈ Σ, hΠ(n) ∈ S(h∈ Π, n ∈ Γ)so that ϕ(hΠ(n)) = ζ(z). We need to prove that there exists a morphism of profinite graphs ψ : S −→ Σ such that ζψ = ϕ and ψ(hΠ(n)) = z.

G ˜ β ϕ˜

ζ α β K Σ Σ δ π α γ ψ

K ψ ζ

ε ι σ ϕ Γ S ϕ p 200 6 Graphs of Pro-C Groups

Since S is a Π-graph, there exists a finite Π-graph such that ϕ factors through , i.e., there exists a morphism of profinite Π-graphs ι : S −→ and a morphism of graphs ϕ : −→ such that ϕ = ϕ ι (see Proposition 2.2.2 and Lemma 2.1.5). Let ζ : ˜ −→ be the universal Galois C-covering of .Fixz˜ ∈ ˜ so that ζ (z)˜ = ι(hΠ(n)); then ϕ ζ (z)˜ = ζ(z).Letϕ˜ : ˜ −→ Σ be the unique morphism of profinite graphs such that ϕ ζ = ζ ϕ˜ and ϕ˜ (z)˜ = z. C By Proposition 3.2.2 π1 ( ) is a normal subgroup of finite index in the profinite H = C group NAut(˜ )(π1 ( )).Let V = H ≤ C V V o ,V π1 . = C ˜ Then V ∈V V 1. Express the profinite π1 ( )-graph as the inverse limit ˜ = \ ˜ lim←− V . V ∈V

Since Σ is ﬁnite, there exists some U ∈ V and a morphism of proﬁnite graphs

δ : Σ = U\˜ −→ Σ such that ϕ˜ = δα, where α : ˜ −→ Σ = U\˜ is the natural projection (see Lemma 2.1.5). Recall that α : ˜ −→ Σ = U\˜ is the universal Galois C-covering = C C of Σ (see Proposition 3.6.1(a)), and U π1 (Σ ). Since U is contained in π1 ( ), the morphism ζ induces a unique morphism of proﬁnite graphs α : Σ −→ such that ζ = α α. Note that α is a pro-C Galois covering with associated group | = C C G(Σ ) π1 ( )/U. Since and π1 ( )/U are ﬁnite, so is Σ . H ≤ C H Observe that NAut(˜ )(π1 (Σ )) since U is normal in . By Proposi- tion 3.2.2, the natural maps

C : H −→ : −→ Φ Aut and ΦΣ NAut(˜ ) π1 Σ Aut Σ are continuous homomorphisms. Since is a Π-graph, there is a continuous homomorphism

μ : Π −→ Aut ,

(if g ∈ Π, μg : −→ is multiplication by g); denote by K the image of this homomorphism:

K = μ(Π) ≤ Aut .

˜ −1 ˜ Put K = Φ (K) ≤ H, and let K = ΦΣ (K). Observe that K ≤ N C (Aut(Σ )), and hence there is a natural continuous homomorphism π1 ( )/U

Φ : K −→ Aut . 6.3 The Standard Graph of a Graph of Pro-C Groups 201 ˜ Furthermore, ΦΦΣ = Φ on K. Then one easily checks the following identities:

Φ k α s = α k s , for all s ∈ Σ ,k ∈ K . (6.10)

Define a map ε : Γ −→ by ε(m) = ι(hΠ(m)) (m ∈ Γ). One checks that ε is a morphism of profinite graphs. Put z = α(z)˜ ∈ Σ ; then α (z ) = ι(hΠ(n)) = ε(n). Since Γ is assumed to be C- simply connected, it is its own universal Galois C-covering; so there exists a unique morphism of profinite graphs γ : Γ −→ Σ such that α γ = ε and γ(n)= z . ∈ K K For m Γ , consider the subgroup γ(m) of -stabilizers of γ(m), and the sub- K K K ≤ K group ε(m) of -stabilizers of ε(m). It follows from (6.10) that Φ( γ(m)) ε(m). We claim that the restriction : K −→ K Φm γ(m) ε(m) K K K˜ ≥ of Φ to γ(m) is an isomorphism onto ε(m). By Proposition 3.2.2(c), C = π1 ( ) Ker(Φ );so

K ≥ Ker(Φ) = G Σ | .

| = C = \ ˜ ∩ Since G(Σ ) π1 ( )/U acts freely on Σ U , we have that Ker(Φ) K = ∈ K γ(m) 1; hence Φm is injective. To check that it is also surjective, ﬁx k ε(m). Let m˜ ∈ ˜ be such that ζ (m)˜ = ε(m). Since ˜ is the universal Galois C-covering of , there exists a unique k˜ ∈ K˜ such that k(˜ m)˜ =˜m and ζ k˜ = kζ . One deduces

˜ ∈ K ˜ = ˜ = that ΦΣ (k) γ(m), and Φ(ΦΣ (k)) Φ (k) k. This proves the claim. Define a map β : G −→ K by β(y) = μ(hν(y)h−1),fory ∈ G(m), m ∈ Γ . Clearly β is continuous and its restriction to each fiber G(m) is a homomorphism, i.e., β is a morphism from G to K. Furthermore, since ν is a specialization, so is β. Note next that β(y) fixes ε(m),fory ∈ G(m), since ι is a morphism of Π-graphs. Define β : G −→ K = −1 ∈ G ∈ by β (y) Φm (β(y)),fory (m), m Γ . Clearly β is a homomorphism when restricted to G(m), for each m ∈ Γ . We claim that β is continuous. To prove this let y ∈ G(m) for a fixed m ∈ Γ . Put k = β(y) and k = β (y). Since K is a finite discrete group, the set

V = π −1 ε−1 ε(m) ∩ β−1(k) 202 6 Graphs of Pro-C Groups is an open neighbourhood of y in G; moreover, β (V ) ={k }. Since K is ﬁnite discrete, this proves the claim. So β is a morphism from G to K . Next we assert that β is a specialization. Note ﬁrst that since γ is a morphism of graphs, γdi(m) = diγ(m)(m∈ Γ, i = 0, 1). Hence

K = K ≥ K (m ∈ Γ, i = 0, 1). γdi (m) di γ(m) γ(m) K It follows that Φm and Φdi (m) coincide on γ(m). So, taking into account that β is a specialization, for each y ∈ G(m) (m ∈ Γ)we have

− − β ∂ (y) = Φ 1 β∂ (y) = Φ 1 β(y) = Φ−1 β(y) = β (y), 0 d0(m) 0 d0(m) m and

β ∂0(y) = β ∂1(y). This proves the assertion. K˜ C K C Since is an extension of π1 ( ) by and these are pro- groups, we deduce that K˜ is a pro-C group; therefore its continuous image K isalsoapro-C group. Let

ρ : Π −→ K be the unique continuous homomorphism such that ρν = β . We claim that

− Φ ρ(g) = μ hgh 1 , for all g ∈ Π. (6.11)

By Corollary 6.2.2, Π is generated by {Π(m)| m ∈ Γ }. Therefore, to verify (6.11) we may assume that g ∈ Π(m);sayg = ν(y),forsomey ∈ G(m). In this case,

= = = −1 = −1 Φ ρ(g) Φ ρν(y) Φ β (y) Φ Φm β(y) μ hgh , proving this claim. Note that

≤ K ∈ ρ Π(m) γ(m) (m Γ), (6.12) = = −1 ∈ K ∈ G since ρν(y) β (y) Φm β(y) γ(m), for every y (m). Deﬁne a map

ψ1 : Γ × Π −→ Σ −1 by ψ1(m, g) = ρ(h g)(γ (m)) (m ∈ Γ, g ∈ Π). Clearly ψ1 is continuous; it is also compatible with the equivalence relation ∼ on Γ × Π,forifm ∈ Γ , g ∈ Π and t ∈ Π(m), then using (6.12),

−1 −1 ψ1(m, gt) = ρ h gt γ(m) = ρ h g ρ(t) γ(m) = ψ1(m, g).

Hence ψ1 induces a continuous map on S = Γ × Π/ ∼

ψ : S −→ Σ 6.3 The Standard Graph of a Graph of Pro-C Groups 203 given by ψ (gΠ(m)) = ρ(h−1g)(γ (m)) (m ∈ Γ, g ∈ Π). Note that

ψ hΠ(n) = γ(n)= z .

Moreover ψ is a morphism of graphs because taking into account that both γ and ρ(h−1g) are morphisms of graphs and that Γ is C-simply connected, we have

−1 ψ di gΠ(m) = ψ gΠ di(m) = ρ h g γ di(m)

−1 = diρ h g γ(m) = diψ gΠ(m) .

We observe next that α ψ = ι, for if m ∈ Γ and t ∈ Π,using(6.11)wehave,

α ψ tΠ(m) = α ρ h−1t γ(m) = Φ ρ h−1t ε(m)

= μ hh−1th−1 ε(m) = μ th−1 ε(m)

= μ th−1 ι hΠ(m) = ι tΠ(m) .

Finally, deﬁne ψ : S −→ Σ to be ψ = δψ . Observe that ψ(hΠ(n)) = z. Since ζδα = ζϕ˜ = ϕ ζ = ϕ α α, and since α is a surjection, we deduce that ζδ= ϕ α . Thus ζψ = ζδψ = ϕ α ψ = ϕ ι = ϕ, as required.

Corollary 6.3.6 (See also Theorem 6.5.2) Assume that C is extension-closed. Let (G,π,Γ) be a graph of pro-C groups over a connected proﬁnite graph Γ , and let S = SC(G,Γ)be the corresponding C-standard graph. Then S is a C-tree.

Proof This is a consequence of the above theorem and Corollary 3.10.2.

Let (G,π,Γ)be a graph of pro-C groups. Choose a continuous 0-section

j : Γ −→ Γ˜ of the universal Galois C-covering ζ : Γ˜ −→ Γ , and let J = j(Γ) be the corresponding 0-transversal. Denote by (ν, ν ) a universal J -specialization of (G,π,Γ) = C G G C in Π Π1 ( ,Γ).Let(β, β ) be a J -specialization of ( ,π,Γ) inapro- group H . Denote by : = C G −→ ϕ Π Π1 ( ,Γ) H 204 6 Graphs of Pro-C Groups the continuous homomorphism induced by (β, β ). Then the natural continuous map

C C ψ : S = S (G,Γ)−→ S (G,Γ,H) deﬁned by

= ∈ ∈ = C G ψ gΠ(m) ϕ(g)H(m) m Γ,g Π Π1 ( ,Γ)

∈ C G ∈ is a morphism of proﬁnite graphs. Indeed, for g Π1 ( ,Γ)and m Γ , one has

ψd0 gΠ(m) = ψ gΠ d0(m) = ϕ(g)H d0(m) = d0 ϕ(g)H(m)

= d0ψ gΠ(m) and

ψd1 gΠ(m) = ψ g ν χ(m) Π d1(m) = ϕ(g) β χ(m) H d1(m)

= d1 ϕ(g)H(m) = d1ψ gΠ(m) .

Using the notation above, we obtain the following characterization of the fundamental group of a graph of groups in terms of its standard graph.

Theorem 6.3.7 Assume that C is an extension-closed pseudovariety of ﬁnite groups. Then the following statements hold. (a) The space of connected components of SC(G,Γ,H) is homeomorphic with the coset space H/Im(ϕ). Consequently, the graph SC(G,Γ,H)is connected if and only if : = C G −→ ϕ Π Π1 ( ,Γ) H is surjective. (b) If ϕ is surjective and the restriction βm = β|G(m) : G(m) −→ H of β to G(m) is injective for every m ∈ Γ , then

C C G =∼ π1 S ( ,Γ,H) Ker(ϕ).

: = C G −→ ϕ Π Π1 ( ,Γ) H

is an isomorphism of proﬁnite groups if and only if SC(G,Γ,H) is C-simply connected. In that case

C C ψ : S = S (G,Γ)−→ S (G,Γ,H)

is an isomorphism of graphs. 6.4 Injective Graphs of Pro-C Groups 205

Proof (a) Let R = Im(ϕ). The composition of natural projections

Γ × H → H → H/R is compatible with the equivalence relation ∼ on the space Γ × H . Hence it induces a continuous map

C ρ S (G,Γ,H)−→ H/R hH(m)→ hR (h ∈ H,m ∈ Γ).

Observe that ρ is a morphism of profinite graphs if H/R is viewed as a profinite graph all of whose elements are vertices. Define [R]= rH(m) r ∈ R,m ∈ Γ .

Then [R]=ψ(S) is a connected proﬁnite subgraph of SC(G,Γ,H), by Proposi- tion 6.3.4. Clearly ρ−1(hR) = h[R] (h ∈ H). We claim that the collection of connected components of SC(G,Γ,H) is {h[R]| h ∈ H }. Clearly C S (G,Γ,H)= h[R]. h∈H On the other hand, h[R] is a connected component, for let C be the connected component of SC(G,Γ,H)containing h[R], and assume that C = h[R]. Then ρ(C)contains at least two different vertices, and thus it is not connected (see Lemma 2.1.9), a contradiction. The map ρ induces a continuous homeomorphism between H/R and the space of connected components of SC(G,Γ,H)obtained by collapsing each of its connected components to a point (see Exercise 2.1.11). (b) Since βm is injective, so is ϕ|Π(m) for every m ∈ Γ . Therefore, Π(m)∩ Ker(ϕ) = 1,(m∈ Γ).

It follows that the action of Ker(ϕ) on S is free (see Lemma 6.3.2(a)). Since ϕ is surjective, so is ψ; hence Ker(ϕ)\S = SC(G,Γ,H), and C ψ : S −→ S (G,Γ,H) is a Galois C-covering with associated group Ker(ϕ). By Theorem 6.3.5, S is C- simply connected; therefore ψ is the universal Galois C-covering of SC(G,Γ,H) C and Ker(ϕ) = π1(S (G,Γ,H))(see Proposition 3.6.1 and Theorem 3.7.1). (c) This follows from (a) and (b).

6.4 Injective Graphs of Pro-C Groups

Let (G,π,Γ) be a graph of pro-C groups over a connected proﬁnite graph Γ , and = C G C let Π Π1 ( ,Γ) be its fundamental pro- group with universal J -specialization 206 6 Graphs of Pro-C Groups

(ν, ν ), where J is a 0-transversal of the universal Galois C-covering ζ : Γ˜ −→ Γ of Γ . Unlike the situation for abstract graphs of groups, the restrictions of ν : G −→ G to each ﬁber G(m) (m ∈ Γ)need not be injective. This can happen even when (G,π,Γ) is a graph of groups over a very simple graph Γ , for example a segment

G G(e) G (v)• •(w)

(cf. Ribes 1971 and RZ, Examples 9.2.9 and 9.2.10 for amalgamated products of profinite groups). We say that a graph of pro-C groups (G,π,Γ) is injective if the restriction of ν to each fiber G(m) (m ∈ Γ)is injective. = C G C As we shall show presently, if Π Π1 ( ,Γ) is the fundamental pro- group of a given graph of pro-C groups (G,π,Γ), one can replace this graph of groups with a naturally defined ‘quotient’ graph of pro-C groups (G˜, π,Γ)˜ , over the C C G˜ = same graph Γ , whose fundamental pro- group Π1 ( ,Γ) coincides with Π C G C Π1 ( ,Γ), and which is an injective graph of pro- groups. Let ϕ : G −→ Γ × Π be the map defined by ϕ(x) = (π(x), ν(x)) (x ∈ G). Since G is compact and ϕ is continuous, its image G˜ = ϕ(G) = (m, y) y ∈ ν G(m) ,m∈ Γ is a compact subset of Γ × Π, and so G˜ is a profinite space. Let π˜ : G˜ −→ Γ be the natural projection. Clearly (G˜, π,Γ)˜ is a sheaf over the space Γ , ϕ is a morphism of sheaves and πϕ˜ = π. Observe that the compactness of G implies that G˜ is a quotient space of G with quotient map ϕ. It follows that the morphism (∂i,di) : (G,π,Γ)−→ (V (G), π, Γ ) induces a morphism

˜ ˜ ˜ (∂i,di) : (G, π,Γ)˜ −→ V(G), π,Γ˜ (i = 0, 1), making G˜ into a graph of pro-C groups over Γ .Letν˜ : G˜ −→ Π be the natural = C G C projection. One easily checks that Π Π1 ( ,Γ) is the fundamental pro- group C G˜ G˜ ˜ ˜ Π1 ( ,Γ) of the graph of groups ( , π,Γ) with universal J -specialization (ν,ν ). Obviously ν˜ is an injection on each fiber G˜(m) (m ∈ Γ), in other words, (G˜, π,Γ)˜ is an injective graph of pro-C groups over Γ with the same fundamental pro-C group as the graph! of groups (G,π,Γ). = C C If G x∈X Gx is a free pro- product of pro- groups continuously indexed by a profinite space X, it can be viewed as the fundamental group of the graph of pro-C groups (G,Γ) constructed in Example 6.2.3(b). This is an injective graph of groups (see Proposition 5.1.6). If G = G1 H G2 is an amalgamated free pro-C product, it can be viewed as = C G a fundamental group G Π1 ( ,Γ) of a graph of groups over a segment, as in Example 6.2.3(d), with G(v) = G1, G(w) = G2 and G(e) = H . This graph of groups is not necessarily injective, as pointed out above; if it is, then one also says that the amalgamated product is proper (cf. RZ, Sect. 9.2). 6.5 Abstract vs Profinite Graphs of Groups 207

Similarly an HNN-extension HNN(G,H,f) is proper if the corresponding graph of groups (see Example 6.2.3(e) with G(v) = G and G(e) = H ) is injective. If (G,π,Γ) is an injective graph of pro-C groups and (ν, ν ) is a universal J - C = C G specialization in its fundamental pro- group Π Π1 ( ,Γ), we often identify G(m) with its image Π(m)= ν(G(m)) in Π.

6.5 Abstract vs Proﬁnite Graphs of Groups

In Sects. 6.2 and 6.3 we have developed the concepts of fundamental group and standard graph of a graph of groups (G,Γ) within the category of pro-C groups. In this section we relate these notions with the corresponding notions for abstract groups when the graph Γ is finite. We have two aims in doing this. On the one hand we will see that in this case we can relax slightly the conditions on the class C C G G C to be able to prove that Π1 ( ,Γ) is well-defined and that S( ,Γ) is a -tree. On the other hand we shall describe how to view the abstract fundamental group, C G in appropriate cases, as a dense subgroup of Π1 ( ,Γ), and the abstract standard graph as a dense subgraph of S(G,Γ); we make this precise below. This connection between the abstract and pro-C notions will be a basic tool in the study of certain properties of abstract groups that we consider in the last chapters of this book. Let (G,Γ)be a graph of abstract groups over a finite connected graph Γ , i.e., for each m ∈ Γ , G(m) is an abstract group such that for each edge e ∈ E(Γ ) there are monomorphisms ∂i : G(e) −→ G(di(e)) (i = 0, 1). abs G We recall the definition of the abstract fundamental group Π1 ( ,Γ) of this graph of groups (cf. Serre 1980, Sect. I.5.1; Dicks and Dunwoody 1989, Defini- tion I.7.3). Choose a maximal subtree T of Γ ; then abs = abs G Π Π1 ( ,Γ) is the abstract group with the following presentation: its generating set consists of the set G(v) ∪ te e ∈ E(Γ ) , v∈V(Γ) and the relations are those of the groups G(v), and in addition,

= −1 ∈ G ∈ = ∈ ∂0(g) te∂1(g)te g (e), e E(Γ ) and te 1ife E(T ). In other words, we have an exact sequence of abstract groups

ϕ abs 1 −→ K1 −→ W1 ∗ Φ −→ Π −→ 1, (6.13) where W1 is the free product = ∗ G W1 (v), v∈V(Γ) 208 6 Graphs of Pro-C Groups

Φ = Φ({te | e ∈ E(Γ )}) is the free group on the set of edges of Γ , and K1 is the normal subgroup of the free product W1 ∗ Φ generated by the set −1 −1 ∈ G ∈ ∪ ∈ ∂0(x) te∂1(x)te x (e), e E(Γ ) te e E(T ) .

abs G It is an easy exercise to check that Π1 ( ,Γ)is characterized by means of a universal property (in the category of abstract groups) analogous to the one described in Example 6.2.3(c) for the fundamental group of a graph of pro-C groups. abs = abs G The group Π Π1 ( ,Γ)is independent of the choice of the maximal subtree T (cf. Serre 1980, Proposition I.20). It is known that the canonical homomorphisms : G → abs G νv (v) Π1 ( ,Γ) are injective (v ∈ V(Γ))(cf. Serre 1980, Sect. I.5.2). Consider now a graph of pro-C groups (G,Γ) over a ﬁnite connected graph Γ . We may think of this also as a graph of abstract groups and form the corresponding exact sequence (6.13). Let M be the collection of all normal subgroups M of W1 ∗Φ such that G(v) ∩ M is open in G(v), for every v ∈ V(Γ) and (W1 ∗ Φ)/M ∈ C. Similarly, let N be the collection of all normal subgroups N of Πabs such that −1 G ∈ abs ∈ C νv (N) is open in (v), for every v V(Γ) and Π /N . Observe that N ={ϕ(M) | M ∈ M}. We think of M and N as systems of fundamental open abs neighbourhoods of 1 deﬁning a topology in W1 ∗ Φ and Π , respectively. One checks that this topology on W1 ∗ Φ induces on each G(v) its natural topology as apro-C group, and it induces on Φ its full pro-C topology. Hence the completion ∗ M C of W1 Φ with respect to! the topology determined by is the free pro- prod- = G C uct W FC, where W v∈V(Γ) (v) and FC is the free pro- group on the set abs abs {te | e ∈ E(Γ )}. Therefore, if K(Π ) denotes the completion of Π with respect to the topology determined by N , we have an exact sequence of pro-C groups

ϕ 1 −→ K −→ W F −→ K Π abs −→ 1, where K is the closure of K1 in W F . This means that

K abs = C G Π Π1 ( ,Γ)

abs G (see Example 6.2.3(c)). Since the deﬁnition of Π1 ( ,Γ) is independent of any C G choice of a maximal tree T of Γ (cf. Serre 1980, Proposition I.20), so is Π1 ( ,Γ). C G Thus we have the following description of Π1 ( ,Γ)when Γ is ﬁnite.

Proposition 6.5.1 Let C be a pseudovariety of finite groups. Let (G,Γ)be a graph C C G of pro- groups over a finite connected graph Γ . Then Π1 ( ,Γ)is the completion abs G C of Π1 ( ,Γ) with respect to the pro- topology determined by the fundamental N abs G system of neighbourhoods of 1 consisting of those N Π1 ( ,Γ) such that −1 G ∈ abs G ∈ C νv (N) is open in (v), for every v V(Γ), and such that Π1 ( ,Γ)/N . 6.5 Abstract vs Profinite Graphs of Groups 209

When Γ is a finite connected graph, this allows us to define the standard graph SC(G,Γ) for any pseudovariety of finite groups C as in Sect. 6.3. One can also sharpen Corollary 6.3.6 and relax slightly the condition on the pseudovariety of finite groups C to guarantee that SC(G,Γ)is a C-tree.

Theorem 6.5.2 Assume that C is a pseudovariety of ﬁnite groups which is closed under extensions with abelian kernel. Let (G,π,Γ) be an injective graph of pro-C groups over a ﬁnite connected graph Γ . Then SC(G,Γ)is a C-tree.

Proof Let Ce be the smallest extension-closed pseudovariety of ﬁnite groups con- C : Ce −→ taining . By Lemma 3.12.1 there is a continuous epimorphism f π1 (Γ ) C e C ˜ −→ π1 (Γ ), and 0-transversals J and J of the universal Galois -covering Γ Γ and Ce-covering Γ˜ e −→ Γ of Γ , respectively, such that

fχe = χ, where : −→ C e : −→ Ce χ Γ π1 (Γ ) and χ Γ π1 (Γ ) are the functions corresponding to J and J e, respectively (see Eq. (3.1) in Sect. 3.4). Let

νe,ν e and ν,ν e G Ce G be a universal J -specialization and J -specialization of ( ,Γ) in Π1 ( ,Γ) and C G Π1 ( ,Γ), respectively. Consider the epimorphism : Ce G −→ C G Ψ Π1 ( ,Γ) Π1 ( ,Γ) induced by the J e-specialization (ν, ν f). Since ν is an injection when restricted to each G(m) (m ∈ Γ)by assumption, so is νe. Therefore Ψ is injective on the sub- e G ∈ = Ce G groups ν ( (m)) (m Γ).SoK Ker(Ψ ) intersects trivially all the Π1 ( ,Γ)- e stabilizers of elements of the standard graph SC (G,Γ) (see Lemma 6.3.2(a)). e Hence K acts freely on SC (G,Γ). It follows from the deﬁnition of standard graph e that SC(G,Γ)= K\SC (G,Γ)and

Ce Ce C Ψ˜ : S (G,Γ)−→ K\S (G,Γ)= S (G,Γ)

e is a Galois Ce-covering with associated group K. By Theorem 6.3.5, SC (G,Γ) is Ce-simply connected and therefore, according to Theorem 3.7.1(a), Ψ˜ is a universal Ce-covering. Thus

= = Ce C G K Ker(Ψ ) π1 S ( ,Γ) . We claim that K is perfect as a profinite group (i.e., it coincides with the closure of its commutator subgroup). Note that a finite abelian group belongs to a pseudovariety of finite groups that is closed under extensions with abelian kernel if and only if 210 6 Graphs of Pro-C Groups for every prime p that divides the order of this group, the cyclic group Cp of order e p belongs to the pseudovariety. Therefore, since Cp ∈ C if and only if Cp ∈ C ,the abelian group K/[K,K] is a pro-C group; thus, since C is closed under extensions Ce G [ ] C with abelian kernel, Π1 ( ,Γ)/ K,K is a pro- group. Let

: Ce G −→ Ce G [ ] ϕ Π1 ( ,Γ) Π1 ( ,Γ)/ K,K Ce G [ ] C C be the natural epimorphism. Since Π1 ( ,Γ)/ K,K is pro- and since π1 (Γ ) C Ce is the largest pro- quotient of π1 (Γ ) (see Proposition 3.12.2), there is a unique continuous homomorphism

e : C −→ Ce G [ ] ϕν π1 (Γ ) Π1 ( ,Γ)/ K,K such that ϕν ef = ϕν e. It follows that

ϕνe, ϕν e

G Ce G [ ] is a J -specialization of ( ,Γ)in Π1 ( ,Γ)/ K,K . This induces a unique continuous homomorphism

: C G −→ Ce G [ ] η Π1 ( ,Γ) Π1 ( ,Γ)/ K,K such that ϕνe = ην and ϕν e = ην . By Proposition 6.2.1(c), η is an epimorphism. Denote by

: Ce G [ ]−→ Ce G [ ] [ ] = C G ψ Π1 ( ,Γ)/ K,K Π1 ( ,Γ)/ K,K / K/ K,K Π1 ( ,Γ) the natural epimorphism.

ϕ Ce G Ce G [ ] Π1 ( ,Γ) Π1 ( ,Γ)/ K,K

Ψ νe η ψ

G ΠC(G,Γ) ν 1

C G = It follows that ψη is an endomorphism of Π1 ( ,Γ)onto itself. Moreover, ψην ν and ψην = ν ; so, by the universal property, ψη is the identity map. It follows that η is an isomorphism. Hence ψ is an isomorphism. This means that K/[K,K]=1, proving the claim. Since π(C) = π(Ce), the result follows from Corollary 3.10.5. 6.5 Abstract vs Proﬁnite Graphs of Groups 211

Next let us start with an arbitrary graph of abstract groups (G,Γ) over a finite connected graph Γ . Denote by Π abs(v) the image of G(v) in Πabs (v ∈ V(Γ)), and abs abs similarly, let Π (e) denote the image of ∂0(G(e)) in Π (e ∈ E(Γ )).LetC be a abs = abs G pseudovariety of finite groups. Assume that the group Π Π1 ( ,Γ) is residually C and denote by Π its pro-C completion. The pro-C topology of Π abs induces apro-C topology on each Πabs(m) (which is not necessarily its full pro-C topology) and hence on G(m) (m ∈ Γ). Define G¯(m) to be the completion of G(m) with respect to this topology. Then the monomorphisms ∂i : G(e) −→ G(di(e)) induce continu- ¯ ¯ ous monomorphisms which we denote again by ∂i : G(e) −→ G(di(e)) (i = 0, 1). We then have a graph of pro-C groups (G¯,Γ) over Γ . The canonical injection G(m) −→ Π abs induces an injection G¯(m) −→ Π(m∈ Γ); furthermore, if we denote by Π(m) the image of G¯(m) on Π under this injection, then Π(m)= Π abs(m), the topological closure of Π abs(m) in Π. Clearly (we are using the presentation for Π abs described at the beginning of this section)

= −1 ∈ G¯ ∈ ∂0(x) te∂1(x)te x (e), e E(Γ ) is in Π (there a certain abuse of notation here, as we are identifying G¯(v) with its image in Π, and similarly we are denoting both the original elements te of the free abs abstract group Φ({te | e ∈ E(Γ )}) and their images in Π ≤ Π). Furthermore, it follows immediately from Proposition 6.5.1 that Π together with the canonical injections G¯(v) → Π is the fundamental pro-C group of the graph of pro-C groups (G¯,Γ) over the ﬁnite graph Γ (note that here we are adopting the point of view explained in Example 6.2.3(c)). We collect this in the following proposition.

Proposition 6.5.3 Let (G,Γ)be a graph of abstract groups over a ﬁnite connected abs = abs G abs graph Γ and let Π Π1 ( ,Γ)be its fundamental group. Assume that Π is residually C and denote by G¯(m) the completion of G(m) with respect to the topology induced on G(m) by the pro-C topology of Π abs. Then (a) (G¯,Γ)is an injective graph of pro-C groups over the ﬁnite graph Γ , and C = C G¯ G¯ C (b) the fundamental pro- group Π Π1 ( ,Γ)of ( ,Γ)is the pro- completion of Π abs.

Associated with the graph of abstract groups (G,Γ), there is an abstract standard graph (or universal covering graph) Sabs = S(G,Γ)which is in fact a tree (cf. Serre 1980, Sect. I.5.3). Recall that (using our notation) the vertices and edges of this tree are V Sabs = Π abs/Π abs(v), and E Sabs = Π abs/Π abs(e) v∈V(Γ) e∈E(Γ ) and its incidence maps

abs abs abs abs d0 gΠ (e) = gΠ d0(e) ,d1 gΠ (e) = gteΠ d1(e) 212 6 Graphs of Pro-C Groups

(g ∈ Π abs,e∈ E(Γ )). Similarly associated with the graph of pro-C groups (G¯,Γ), there is a C-standard graph S = SC(G¯,Γ) (see Example 6.2.3(c)). Assume further that the pseudovariety of ﬁnite groups C is closed under extensions with abelian kernel. Then according to Theorem 6.5.2, S = SC(G¯,Γ) is in fact a C-tree, with spaces of vertices and edges V(S)= Π/Π(v) and E(S) = Π/Π(e) v∈V(Γ) e∈E(Γ ) and with incidence maps

d0 gΠ(e) = gΠ d0(e) ,d1 gΠ(e) = gteΠ d1(e) g ∈ Π,e ∈ E(Γ ) . It follows that there is a well-deﬁned natural morphism of graphs

ϕ : Sabs −→ S which on vertices and edges is gΠabs(v) → gΠ(v), gΠ abs(e) → gΠ(e) g ∈ Π abs,v∈ V(Γ),e∈ E(Γ ) .

For this morphism ϕ to be injective one needs that Π abs(v) = Π abs ∩ Π(v) and Π abs(e) = Π abs ∩ Π(e) (v ∈ V(Γ),e∈ E(Γ )), i.e., that each Π abs(m) (m ∈ Γ)be closed in the pro-C topology of Π abs (m ∈ Γ). If this is the case, we think of ϕ as an embedding of graphs. Note that its image is dense in S. We collect this in the following proposition.

Proposition 6.5.4 We continue with the hypotheses and notation of Proposi- tion 6.5.1. Furthermore, we assume that the pseudovariety of ﬁnite groups C is closed under extensions with abelian kernel and that Π abs(m) is closed in the pro- C topology of Π abs, for every m ∈ Γ . Then the standard (or universal covering) tree Sabs = S(G,Γ) of the graph of groups (G,Γ) is canonically embedded in the C-standard C-tree S = SC(G¯,Γ) of the graph of pro-C groups (G¯,Γ), and Sabs is dense in S.

We end this section with an application of the above results to the case of a graph of groups (G,Γ) over a finite graph Γ such that each G(m) is a finite group in C (m ∈ Γ).IfΠ abs(G,Γ) is residually C, then the topology on G(m) induced from the pro-C topology of Π abs(G,Γ) is the discrete topology (i.e., its own pro-C topology and G¯(m) = G(m), for all m ∈ Γ ). Therefore in this case the hypotheses of Proposition 6.5.3 hold, and so (G,Γ) is an injective graph of pro-C groups. On the other hand, since each Π abs(m) is a finite group, it is closed in the pro-C topology of Π abs(G,Γ), so that the hypotheses of Proposition 6.5.4 also hold. Hence the first two parts of the following result hold.

Proposition 6.5.5 Let (G,Γ)be a graph of groups over a finite graph Γ such that each G(m) is a finite group in C (m ∈ Γ). Assume that Π abs(G,Γ) is residually C. Then 6.6 Action of a Pro-C Group on a Profinite Graph with Finite Quotient 213

G C = C G C (a) ( ,Γ) is an injective graph of pro- groups and Π Π1 ( ,Γ) is the pro- abs = abs G completion of Π Π1 ( ,Γ); (b) if C is closed under extensions with abelian kernel, then Sabs = Sabs(G,Γ) is canonically embedded in the C-tree S = SC(G,Γ)and Sabs is dense in S; (c) . abs S = αιS , abs αι∈Π/Π

abs abs where the αι form a left transversal of Π in Π. In particular, S = Sabs(G,Γ) is an abstract connected component of S = SC(G,Γ) considered as an abstract graph. abs Proof It remains to prove (c). Since Π = . ∈ abs α Π , we have that S = αι Π/Π ι abs abs ΠS = abs α S . To show that this union is disjoint, it sufﬁces to prove αι∈Π/Π ι that if α ∈ Π and αv ∈ Sabs, where v ∈ Sabs, then α ∈ Π abs. Note that since in this case G(m) is ﬁnite, Π abs(m) = Π(m), for every m ∈ Γ . Say v = aΠ(m) = aΠabs(m),forsomem ∈ Γ , where a ∈ Π abs. Then αaΠabs(m) = bΠabs(m),for some b ∈ Π abs. Hence b−1αa ∈ Π abs(m) ≤ Π abs. Thus α ∈ Π abs, as desired.

If in the above proposition C is the class of all ﬁnite groups, one can simplify the assumptions since then Π abs(G,Γ) is automatically residually ﬁnite (cf. Serre 1980, Part II, Proposition 11). Then the above proposition can be written as follows.

Proposition 6.5.6 Let C be the pseudovariety of all finite groups. Let (G,Γ) be a graph of groups over a finite graph Γ such that each G(m) is a finite group (m ∈ Γ). Then G = C G (a) ( ,Γ)is an injective graph of profinite groups and Π Π1 ( ,Γ)is the profi- abs = abs G nite completion of Π Π1 ( ,Γ); (b) Sabs = S(G,Γ) is canonically embedded in S = SC(G,Γ) and Sabs is dense in S; (c) Sabs = S(G,Γ) is an abstract connected component of S = SC(G,Γ) considered as an abstract graph.

6.6 Action of a Pro-C GrouponaProﬁniteGraphwithFinite Quotient

If G is an abstract group that acts on an abstract tree Σ, the theory of BassÐSerre for abstract groups and abstract graphs gives a natural and often very useful way of describing the structure of G in terms of G-stabilizers of certain vertices and edges of Σ. We recall brieﬂy how this is done. Consider the (connected) quotient graph Γ = G\Σ; then there exists a spanning tree T of Γ and the natural epimorphism Σ −→ Γ = G\Σ admits (in our terminology) a fundamental 0-section 214 6 Graphs of Pro-C Groups

σ : Γ −→ Σ such that σ|T is an isomorphism of trees onto its image. One can define a natural graph of groups (G,Γ)such that G(m) = Gσ(m),theG-stabilizer of σ(m) ∈ abs G (m Γ). Then one has that G is the (abstract) fundamental group Π1 ( ,Γ).For details one may consult Serre (1980), Theorem I.13. The situation for pro-C groups that act on C-trees is necessarily more complex. For example, a nontrivial p-Sylow subgroup G of a nonabelian free profinite group F(Y,∗) on a pointed space (Y, ∗) acts naturally on the profinite tree Γ(F,Y)(see Theorem 2.5.3); but obviously G is not the fundamental profinite group of any graph of groups whose fibers are G-stabilizers of elements of Γ(F,Y): indeed such stabilizers are trivial and G would have to be isomorphic to the fundamental profinite group of a profinite graph (see Proposition 6.2.1(c)); in turn, this fundamental group is either a free profinite group, if is finite, or it would have quotients which are free profinite groups of arbitrary large finite rank (see Propositions 3.5.3(b) and 3.3.2(b)), and either of these alternatives is absurd since G is a pro-p group. When trying to adapt the program described above in the case of abstract groups to the action of a pro-C group on a C-tree (or more generally, on a C-simply connected profinite graph), one encounters several difficulties. Let G be a pro-C group that acts continuously on a connected C-simply connected graph Σ, and let Γ = G\Σ be the corresponding quotient graph. The first obstacle is that Γ need not have a spanning C-simply connected profinite subgraph (see Example 3.4.1). But even if one assumes the existence of such a subgraph, there is no general procedure to construct a fundamental 0-section of the natural quotient morphism Σ −→ Γ (see Example 3.4.2). These difficulties disappear when Γ is a finite graph, and then the above program can be carried out smoothly. Indeed, in this case there exists a maximal subtree T (which is automatically a spanning subtree) of Γ . Construct a fundamental 0-section j : Γ −→ Γ˜ of the quotient morphism Γ˜ −→ Γ lifting T , and a fundamental 0- section δ : Γ −→ Σ of the quotient morphism Σ −→ Γ lifting T . Observe that d1δ(m) and δd1(m) areinthesameG-orbit; choose tm ∈ G such that

tmδd1(m) = d1δ(m), = ∈ ∈ ∈ C and note that tm 1, for m T . Similarly, for each m Γ ,letχ(m) π1 (Γ ) be such that

χ(m)jd1(m) = d1j(m) (see Eq. (3.1) in Sect. 3.4). We deﬁne a graph of pro-C groups (G,Γ)as follows: for m ∈ Γ , put

G(m) = Gδ(m), the G-stabilizer of the element δ(m) under the action of G. The incidence morphisms ∂0 and ∂1 are deﬁned as follows: on each ﬁber G(m), ∂0 is just the inclusion map

G(m) → G d0(m) , 6.6 Action of a Pro-C Group on a Proﬁnite Graph with Finite Quotient 215 and ∂1 is the composition of an inclusion and conjugation by tm:

−1 G(m) = Gδ(m) −→ G d1(m) = Gδd (m) = G −1 = t Gd δ(m)tm, 1 tm d1δ(m) m 1

→ −1 ∈ G x tm xtm x (m) . Let : C −→ γ π1 (Γ ) G be the unique continuous homomorphism such that γ χ(m) = tm (m ∈ Γ) (see Corollary 3.5.4). On the other hand, the inclusion maps

G(m) = Gδ(m) → G(m∈ Γ) determine a morphism γ : G −→ G. One checks that (γ, γ ) is a J -specialization of G in G, where J = j(Γ). Hence it induces a continuous homomorphism : C G −→ ϕ Π1 ( ,Γ) G, such that ϕν = γ and ϕν = γ , where (ν, ν ) is the universal J -specialization of G C G ( ,π,Γ) in Π1 ( ,Γ). We remark that this implies that ν is an injection on each G(m) (m ∈ Γ), i.e., that (G,π,Γ)is an injective graph of groups.

Theorem 6.6.1 Let C be extension-closed. Suppose that a pro-C group G acts on a C-simply connected profinite graph Σ so that the quotient graph Γ = G\Σ is finite. Construct a graph of pro-C groups (G,Γ) over Γ as above. Then the homomorphism : C G −→ ϕ Π1 ( ,Γ) G defined above is an isomorphism of profinite groups. Moreover, Σ is isomorphic to the standard graph SC(G,Γ)of the graph of pro-C groups (G,Γ).

Proof Deﬁne a map Ψ : Γ × G −→ Σ by Ψ(m,g)= gδ(m) (m ∈ Γ). Clearly Ψ is continuous and onto. It induces a continuous map C Ψ : S (G,Γ,G)= . G/G(m) −→ Σ, m∈Γ given by Ψ (gG(m)) = gδ(m), which is a bijection, and so a homeomorphism. We claim that Ψ is an isomorphism of proﬁnite graphs. To see this it remains to check that it is a morphism of graphs: if g ∈ G and m ∈ Γ ,wehave

Ψ d0 gG(m) = Ψ gG d0(m) = gδ d0(m) = d0 gδ(m) = d0Ψ gG(m) , 216 6 Graphs of Pro-C Groups and

Ψ d1 gG(m) = Ψ gγ χ(m)G d1(m) = Ψ gtmG d1(m) = gtm δd1(m)

= g d1δ(m) = d1 gδ(m) = d1Ψ gG(m) .

This proves the claim. Since Σ is C-simply connected, so is SC(G,Γ,G). Therefore by Theorem 6.3.7, : C G −→ C G the homomorphism ϕ Π1 ( ,Γ) G is an isomorphism, and S ( ,Γ,G) is isomorphic to SC(G,Γ). It follows that Σ and SC(G,Γ)are isomorphic.

Corollary 6.6.2 Let C be extension-closed. Suppose that a pro-C group G acts freely on a C-simply connected proﬁnite graph Σ so that the quotient graph Γ = \ C C G Σ is ﬁnite. Then G is the free pro- group π1 (Γ ).

= C G G = ∈ Proof By Theorem 6.6.1, G Π1 ( ,Γ), (m) 1(m Γ ), since G acts freely = C C on Σ. Hence (see Corollary 6.2.5) G π1 (Γ ); this is the free pro- group with basis Γ − T , where T is a maximal subtree of Γ (see Proposition 3.5.3(b)).

As we saw in Theorem 4.1.2, a pro-C group that acts freely on a C-tree is necessarily projective. One can ensure that G is free pro-C if the ﬁnite groups in C are solvable:

Corollary 6.6.3 Let C be an extension-closed pseudovariety of ﬁnite solvable groups. Suppose that a pro-C group G acts freely on a C-tree Σ so that the quo- = \ C C tient graph Γ G Σ is ﬁnite. Then G is the free pro- group π1 (Γ ).

Proof Since C consists of ﬁnite solvable groups, Σ is C-simply connected (see Proposition 3.10.4(b)); so the result is a special case of the above corollary.

6.7 Notes, Comments and Further Reading: Part I

The beginning of the theory of profinite graphs appears in Gildenhuys and Ribes (1978) where the name boolean graph is used. The aim was to develop a parallel to the BassÐSerre theory of abstract groups acting on abstract trees (cf. Serre 1977, 1980) for profinite groups and applications to abstract groups. Our notion of profinite graph corresponds to the concept of ‘orientation’ for an abstract graph in the treatment of Serre (1980);seealsoDicks(1980), Dicks and Dunwoody (1989); for a slightly different notion of graph (with a combinatorial point of view) see Tits (1970), or still Tits (1977) (the approach in this paper is rather geometric, with trees being certain types of metric spaces). We put the emphasis on the geometric aspect of the theory, for example when dealing with the concept of ‘chain’ [v,w] determined by two points v and w in a π-tree. The Cayley 6.7 Notes, Comments and Further Reading: Part I 217 graph of a group with respect to a subset in the abstract case has a natural orientation. Our notion of Cayley graph Γ(G,X)differs slightly from the usual definition in the abstract setting; our motivation is the need to take account of the fact that every profinite group admits a set of generators converging to 1, and the closure of such generating set contains 1. As we have pointed out, both definitions coincide when 1 ∈/ X, and this is important in the last chapters of this book when we apply the theory of profinite graphs to obtain properties of abstract groups; there it is crucial to study the interplay between certain abstract groups G and their profinite completions, Gˆ , and also between graphs associated to G and Gˆ . This relationship with abstract graphs is again our motivation for introducing the terminology ‘qmorphism’ vs ‘morphism’; we want to accommodate the fact that for abstract graphs it is traditionally assumed that a ‘morphism’ sends edges to edges. A refined version of the concept of π-tree was introduced in Zalesskii and Mel’nikov (1988), and this is the notion that we are using in this book. Instead of the terms C-tree, π-tree or p-tree one often finds in the literature the terms pro-C tree, pro-π tree or pro-p tree. In this book we are using the former terminology to avoid the misleading impression that a C-tree is the inverse limit of finite trees (see Example 2.4.5). Nevertheless, as we have indicated, for convenience we use the term ‘profinite tree’ to refer to a π-tree, when π is the class of all prime numbers. Lemma 2.5.2 and Theorem 2.5.3 were proved in one direction in Gildenhuys and Ribes (1978). The converse direction was proved by Almeida and Weil (1994). The universal C-covering graph of a connected profinite graph was defined in Gildenhuys and Ribes (1978), where it is proved that it is a C-tree under certain conditions; the proof that we present here of Theorem 3.10.1 is essentially taken from that paper. Also in that paper freeness of a pro-C group G on a pointed profinite space (X, ∗) is shown to be equivalent to the Cayley graph Γ(G,X) being a C- tree, but only when the class C consists of solvable groups (Corollary 3.11.2). This motivated Zalesskii (1989) to introduce the important distinction between C-tree and C-simply connected profinite graph; Galois coverings of profinite groups were first studied in that paper. These two concepts coincide if the pseudovariety C consists of solvable groups (Corollary 3.10.2 and Proposition 3.10.4(b)). In general one can then characterize freeness of a pro-C group G on a pointed subspace (X, ∗) in terms of the C-simply connectedness of Γ(G,X)(Theorem 3.11.1). The presentation in Sect. 4.1 is based on results of Zalesskii and Mel’nikov (1988). For an analogue of Theorem 4.1.8, in the context of abstract groups and abstract trees, see Serre (1980), Sect. 6, Theorem 15. The (simpler) argument of Serre for the abstract case uses the fact that G\T is a tree and so it can be lifted to T , something that is not guaranteed in the profinite case. The presentation in Sect. 4.2 is based on results of Zalesskii (1990). The first results of this type appear in Herfort and Ribes (1989a), where a description of the solvable subgroups of free products of profinite groups is given using profinite Frobenius groups; as we see in Chap. 5, any such a free product acts naturally on a tree canonically associated with it. See also Herfort and Ribes (1989b). It is also shown in these papers and in Guralnick and Haran (2011) that the alternatives in Theorem 4.2.11 actually arise already if one considers subgroups of free products of profinite groups. 218 6 Graphs of Pro-C Groups

The first attempt to define a free product of profinite groups indexed by a space that is not strictly discrete appears in Neukirch (1971). In Gildenhuys and Ribes (1973) there appears for the first time the idea of a collection of pro-C groups indexed by a topological space T , the concept of ‘sheaf’ (a version of it there is called ‘étale space’) and the corresponding free pro-C product. However it is done in a very restrictive case, namely when indexing is ‘almost’ locally constant. This concept of free product is not sufficiently general to describe certain natural groups; it is only sufficiently good to handle the structure of open subgroups of that type of free product, or certain particular closed subgroups, for example, the structure of the normal closure of A in a free pro-C product of the form A B, where A and B are pro-C groups. Haran (1987) and Mel’nikov (1989) independently expand these ideas and develop rather general approaches to free products of profinite groups indexed by a profinite space; their aim was to be able to describe the structure of at least certain closed subgroups of free pro-p products of pro-p groups (see Sect. 9.6). The papers of Haran and Mel’nikov obtain similar group theoretic results. In Chap. 5 we have adopted the rather elegant viewpoint of Mel’nikov. The term ‘sheaf’ that we adopt here appears to be the most commonly used in the literature on this topic now. The alternatives could be ‘bundle’ or ‘étale space’, but none of these three names really reflect exactly, in their traditional meanings, the concept that is developed here. The étale topology for Subgp(G) defined in Sect. 5.2 appears in Haran, Jarden and Pop (2009), where free profinite products are used to describe certain absolute Galois groups. A version of Corollary 5.3.3 appears in Gildenhuys and Lim (1972). Theorem 5.3.4 appears in Ribes (1990). The results of Sect. 5.4 are taken from Ribes (2008). Versions of Theorem 5.5.6 appear in Haran and Lubotzky (1985) and Herfort and Ribes (1989b). The first sections in Chap. 6 are based on Zalesskii and Melnikov (1989). To C G prove that the fundamental group Π1 ( ,Γ) is well-defined we have assumed in Theorem 6.2.4 that the pseudovariety of finite groups C is extension-closed. This is not always necessary. If Γ is a finite graph, we have indicated in Sect. 6.5 that C G C Π1 ( ,Γ) is a pro- completion of the corresponding abstract fundamental group, and this does not require special conditions on the pseudovariety C.

Abstract Graph of Finite Groups (G,Γ)over an Inﬁnite Graph Γ

As we shall see in Chaps. 8, 14 and 15, Propositions 6.5.3Ð6.5.6 can be used very effectively in the study of certain finitely generated abstract groups. In these propositions it is always assumed that the graph Γ in the graph of groups (G,Γ)is finite. The main reason for this assumption is that then Γ is automatically a profinite graph and so (G¯,Γ)is automatically a profinite graph of groups, and one has a natural way abs = abs G = G¯ abs = abs G of relating Π Π1 ( ,Γ) with Π Π1( ,Γ) and S S ( ,Γ) with S = S(G¯,Γ). With a view to possible applications to abstract groups which are not necessarily finitely generated, one can seek generalizations of Propositions 6.5.3Ð 6.5.6 when the abstract graph Γ is infinite. To be specific, we state one of several possible questions. 6.7 Notes, Comments and Further Reading: Part I 219

Open Question 6.7.1 Let (G,Γ) be an abstract graph of finite groups over an abstract graph Γ , which is infinite. In analogy with Proposition 6.5.6, is there a way of defining a profinite graph Γ¯ that contains Γ as a dense subgraph, and a profinite graph of finite groups (G , Γ)¯ , with G (m) = G(m), whenever m ∈ Γ , such that ¯ (a) the profinite fundamental group Π1(G , Γ) is ‘a’ profinite completion of abs G Π1 ( ,Γ), where it is embedded, and (b) Sabs(G,Γ) is densely embedded in S(G , Γ)¯ in such a way that the ac- G ¯ G ¯ abs G tion of Π1( , Γ) on S( , Γ) induces the natural action of Π1 ( ,Γ) on Sabs(G,Γ)? Part II Applications to Profinite Groups

In the next four chapters we use results of Part I to obtain structural information about profinite groups and their actions on C-trees. Chapter 7 deals with the de- C G scription of subgroups of the fundamental group Π1 ( ,Γ)of an injective graph of groups (G,Γ) in terms of the vertex and edge groups G(m). As in the BassÐSerre C G C theory, this is done via a study of the action of Π1 ( ,Γ) on the standard -tree SC(G,Γ) associated with the graph of pro-C groups (G,Γ). In particular, an analogue of the Kurosh subgroup theorem for open subgroups of free pro-C products is proved in Sect. 7.3. Chapter 8 is concerned with minimal R-invariant subtrees of a tree T on which a group R acts: this is studied both when R and T are abstract and profinite. The cases of main interest are those of a free group and the corresponding Cayley graph, a finitely generated subgroup of a free product of groups and the corresponding tree, or a cyclic subgroup of a fundamental group of a graph of groups and the corresponding standard tree. This study provides useful tools when investigating properties such as conjugacy separability in abstract groups. Chapter 9 extends some classical homological algebra to describe the behaviour ⊗−ˆ Λ − of the complete tensor product M , or more generally the functors Torn (M, ), when applied to a direct sum of a family of submodules continuously indexed by a profinite space. It includes the existence of a MayerÐVietoris exact sequence associated with the fundamental group of a graph of profinite groups. It contains a homological characterization of when a pro-p group is the free pro-p product of a family of subgroups continuously indexed by a profinite space. There is also a study of pro-p groups acting continuously on C-trees and, as a consequence, a pro- p version of the Kurosh subgroup theorem describing a countably generated pro-p subgroup of a free pro-C product of pro-C groups. Chapter 10 contains a result of Serre that asserts that torsion-free virtually free pro-p groups are free pro-p, as well as an extension of Scheiderer of this result when the group contains torsion. It also includes an example of a subgroup of a free product of pro-p groups which does not admit a description along the lines of the classical Kurosh subgroup theorem. The last part of this chapter deals with the 222 subgroup of fixed points FixF (ψ) of an automorphism ψ of a free pro-p group F : if the order of ψ is a finite power of p, the rank of that subgroup is finite, and if the order of ψ is prime to p, its rank is infinite. Chapter 7 Subgroups of Fundamental Groups of Graphs of Groups

Throughout the chapter we consider only injective graphs of pro-C groups (G,Γ) over a proﬁnite graph Γ . We are interested in the description of closed subgroups of C = C G the fundamental pro- groups Π Π1 ( ,Γ)of such graphs of groups in terms of the vertex and edge groups G(m). We often identify G(m) with its image Π(m)in Π (m ∈ Γ). Theorem 6.5.2 allows us to apply the results of Chap. 4 to obtain structural information about subgroups of free products, of proper free products with amalgamations, of proper HNN-extensions and, more generally, of fundamental groups of injective graphs of pro-C groups.

7.1 Subgroups

In this section we shall assume that C is an extension-closed pseudovariety of ﬁnite groups.

Proposition 7.1.1 Let H be a closed subgroup of the fundamental pro-C group = C G C G Π Π1 ( ,Γ) of an injective graph of pro- groups ( ,Γ) and let M be the closed subgroup of H generated by all intersections H ∩ gΠ(v)g−1, ∈ ∈ C G with v V(Γ)and g Π1 ( ,Γ). Then M is normal in H and H/M is projective. = C G C G Proof consider the natural action of Π Π1 ( ,Γ)on S ( ,Γ). Then H also acts on SC(G,Γ). By Corollary 6.3.6, SC(G,Γ)is a C-tree. Then the result follows from Corollary 4.1.3.

Theorem 7.1.2 Let K be a ﬁnite subgroup of the fundamental pro-C group Π = C G C G Π1 ( ,Γ)of an injective graph of pro- groups ( ,Γ). Then K ≤ gΠ(v)g−1, ∈ ∈ C G for some v V(Γ)and g Π1 ( ,Γ).

© Springer International Publishing AG 2017 223 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_7 224 7 Subgroups of Fundamental Groups of Graphs of Groups

Proof This follows from Corollary 6.3.6 and Theorem 4.1.8.

Corollary 7.1.3 (a) Let G beapro-C group and assume that it can be expressed as a free pro-C product

G = Gx x∈X

of closed subgroups Gx continuously indexed by a profinite space X. Let K be −1 a finite subgroup of G. Then K ≤ gGx g , for some g ∈ G and some x ∈ X. (b) Let K be a finite subgroup of a proper amalgamated free pro-C product

G = G1 H G2 −1 of pro-C groups. Then K ≤ gGig , for some g ∈ G and i = 1 or 2. (c) Let K be a ﬁnite subgroup of a proper pro-C HNN-extension G = HNN(H,A,f), where H is a pro-C group, A is a closed subgroup of H and f is an isomorphism of proﬁnite groups from A onto a closed subgroup of H . Then K ≤ gHg−1, for some g ∈ G.

= C G C Theorem 7.1.4 Let Π Π1 ( ,Γ)be the fundamental pro- group of an injective graph of pro-C groups (G,Γ) over a profinite graph Γ such that V(Γ) is clopen (e.g., Γ is finite). Suppose that v,w ∈ V(Γ), g ∈ Π and either v = w or g/∈ Π(v). Then Π(v)∩ gΠ(w)g−1 ≤ bΠ(e)b−1, for some edge e ∈ E(Γ ) incident with v, and some b ∈ Π. Furthermore, there exists an e ∈ E(Γ ) incident with v, and a c ∈ Π(v), such that either − − Π(v)∩ gΠ(w)g 1 ≤ cΠ(e)c 1, (7.1) or Π(v)∩ gΠ(w)g−1 ≤ cχ(e)−1Π(e)χ(e)c−1, : −→ C where χ Γ π1 (Γ ) is the function considered in Eq.(3.1) in Sect. 3.4. Conse- quently, if Γ is C-simply connected, the first alternative (7.1) is always valid.

= C G C = C G Proof Consider the action of Π Π1 ( ,Γ) on the -tree S S ( ,Γ) (see Corollary 6.3.6). If m ∈ Γ , put m¯ = 1Π(m) ∈ S. Note that Πm¯ = Π(m) and −1 Πxm¯ = xΠ(m)x (x ∈ Π). By assumption the set of edges E(Γ ) is closed in Γ ; therefore E(S) is compact; hence so is E([¯v,gw¯ ]) =[¯v,gw¯ ]∩E(S). By Proposi- tion 2.1.6(c), there exists some e ∈ E([¯v,gw¯ ]) such that one of its end points is v¯, i.e., di(e ) =¯v, where e = bΠ(e),forsomee ∈ E(Γ ), b ∈ Π and i = 0or1.By Corollary 4.1.6, Πv¯ ∩ Πgw¯ ≤ Πe . It follows that −1 −1 Π(v)∩ gΠ(w)g ≤ Πe = bΠ(e)b . 7.1 Subgroups 225

Now, if d0(e ) =¯v, then 1Π(v)= bΠ(d0(e)) = bΠ(v); hence b ∈ Π(v); so we can choose c = b. On the other hand, if d1(e ) =¯v, then 1Π(v) = bχ(e)Π(d1(e)) = bχ(e)Π(v); hence bχ(e) ∈ Π(v). Putting c = bχ(e), the last alternative follows. Finally, observe that χ(m)= 1 for all m ∈ Γ if Γ is C-simply connected.

Corollary 7.1.5 (a) Let G beapro-C group and assume that it can be expressed as a free pro-C product

G = Gx x∈X

of closed subgroups Gx continuously indexed by a proﬁnite space X. Suppose that x,y ∈ X and g ∈ G. Then −1 Gx ∩ gGyg = 1,

whenever x = y or g/∈ Gx . (b) Let G = G1 H G2 be a proper amalgamated free pro-C product of pro-C groups. Then −1 −1 Gi ∩ gGj g ≤ bHb ,

for some b ∈ Gi , whenever i = j or g/∈ Gi (i, j ∈{1, 2}). (c) Let G = HNN(H,A,f)be a proper pro-C HNN-extension. Then H ∩ gHg−1 ≤ bAb−1 or H ∩ gHg−1 ≤ bf (A)b−1, for some b ∈ H .

Corollary 7.1.6 = C G C (a) Let Π Π1 ( ,Γ) be the fundamental pro- group of an injective graph of pro-C groups (G,Γ)where Γ is C-simply connected. Let v ∈ V(Γ). If Π(v)= Π(e), for every edge e of Γ incident with v, then Π(v) is its own normalizer in Π. (b) Let G beafreepro-C product

G = Gx x∈X

of closed nontrivial subgroups indexed by a proﬁnite space X. Then Gx is its own normalizer in G (x ∈ X). (c) Let G be a proper amalgamated free pro-C product G = G1 H G2 of pro-C groups such that G1 = H = G2. Then Gi is its own normalizer in G (i = 1, 2). (d) Let G be a proper pro-C HNN-extension G = HNN(H,A,f) such that A = H = f(A). Then H is its own normalizer in G.

Proof (a) Assume that NΠ (Π(v)) > Π(v) and let g ∈ NΠ (Π(v)) − Π(v). Then by Theorem 7.1.4 we get Π(v)= Π(v)∩ gΠ(v)g−1 ≤ bΠ(e)b−1, 226 7 Subgroups of Fundamental Groups of Graphs of Groups where e is an edge in Γ incident with v, and b ∈ Π(v). Hence Π(v) ≤ Π(e), and so Π(v)= Π(e), contradicting our assumptions. Thus NΠ (Π(v)) = Π(v). Parts (b) and (c) are special cases of (a) (see Example 6.2.3(b), (d)). (d) Assume that NΠ (H ) > H , and let g ∈ NΠ (H ) − H . By Corollary 7.1.5(c), H ∩ gHg−1 ≤ bAb˜ −1, where A˜ = A or f(A), and b ∈ H . It follows that H ≤ A˜ and so H = A˜, a contradiction.

In the next theorem and its corollary, if σ is a nonempty set of primes, we shall denote by C(σ ) the smallest extension-closed pseudovariety of ﬁnite groups containing Cp, for every p ∈ σ .

Theorem 7.1.7 Let H be a closed subgroup of the fundamental pro-C group Π = C G C G Π1 ( ,Γ) of an injective graph of pro- groups ( ,Γ). Then one of the following assertions holds: ≤ −1 = C G (a) H gΠ(v)g , where Π(v) is a vertex subgroup of Π Π1 ( ,Γ), and g is some element of Π. (b) For some p ∈ π(C), H has a nonabelian free pro-p subgroup P such that P ∩ −1 = = C G gΠ(v)g 1 for every vertex subgroup Π(v) of Π Π1 ( ,Γ) and every g ∈ Π. = C G (c) There exists an edge subgroup Π(e) of the group Π Π1 ( ,Γ) and an ele- −1 ment g ∈ Π such that the subgroup H0 = H ∩ gΠ(e)g is normal in H , and H/H0 is solvable and isomorphic to one of the following groups:

(c1) a projective group of the form Z ˆ Z ˆ , where σ,ρ ⊆ π(C) with σ ∩ρ =∅; σ ρ ∼ (c2) a proﬁnite Frobenius group of the form Zσˆ C, σ ⊆ π(C), C = Cn, with isolated subgroup C; ∼ C(σ ) (c3) an inﬁnite dihedral pro-σ group of the form Z ˆ C = C C , where ∼ σ 2 2 2 ∈ σ ⊆ π(C), C = C2 and C acts on Zσˆ by inversion. = C G C = C G Proof Consider the action of Π Π1 ( ,Γ)on the -tree S S ( ,Γ)(see The- orem 6.5.2). Since H is a subgroup of G, it acts naturally on S as well. The assertions of this theorem follow then from Theorem 4.2.11: just observe that the vertex and edge stabilizers of S under the action of H have the form H ∩ gΠ(v)g−1 and H ∩ gΠ(e)g−1, respectively (v ∈ V(Γ),e∈ E(Γ ),g ∈ G).

We present next a corollary making Theorem 7.1.7 explicit for free pro-C products. Similar results can be stated for amalgamated free pro-C products and pro-C HNN-extensions.

Corollary 7.1.8 Let

G = Gx x∈X beafreepro-C product of closed subgroups Gx continuously indexed by a proﬁnite space X, and let H be a closed subgroup of G. Then one of the following assertions holds: 7.2 Normal Subgroups 227

−1 (a) H ≤ gGx g , for some g ∈ G and some x ∈ X. (b) For some p ∈ π(C), H has a nonabelian free pro-p subgroup P such that P ∩ −1 gGxg = 1, for every x ∈ X and every g ∈ G. (c) H is solvable and isomorphic to one of the following groups:

(c1) a projective group of the form Z ˆ Z ˆ , where σ,ρ ⊆ π with σ ∩ ρ =∅; σ ρ ∼ (c2) a proﬁnite Frobenius group of the form Zσˆ C, σ ⊆ π, C = Cn, with isolated group C; ∼ C(σ ) (c3) an inﬁnite dihedral pro-σ group of the form Z ˆ C = C C , where ∼ σ 2 2 2 ∈ σ ⊆ π, C = C2, and C acts on Zσˆ by inversion.

Proof This follows from Theorem 7.1.7 and the fact that a free pro-C product can be expressed as the fundamental group of a graph of pro-C groups all of whose edge groups are trivial (see Example 6.2.3(b)).

Remark 7.1.9 The groups described in all the cases of Corollary 7.1.8 do actually appear as closed subgroups of free pro-C products of pro-C groups (cf. Herfort and Ribes 1989a and Guralnick and Haran 2011).

7.2 Normal Subgroups

In this section we shall assume that C is a pseudovariety of ﬁnite groups closed under extensions with abelian kernel, unless it is otherwise indicated.

Let (G,Γ)be an injective graph of pro-C groups over a finite connected graph Γ . Suppose e ∈ E(Γ ) is an edge such that d0(e) = d1(e) and ∂i(G(e)) = G(di(e)) for either i = 0ori = 1. Define a new graph Γ = Γ/{e} obtained by collapsing the subgraph {e,d0(e), d1(e)} to a vertex ve (see Example 2.1.2). Let (G ,Γ ) be the graph of pro-C groups over Γ , where G (m) = G(m) if m ∈ Γ,m= e,d0(e), d1(e), and G (v ) = G(d − (e)), and where the morphisms e 1 i : G −→ G ∂i ,di ,Γ V ,V Γ are defined naturally from (∂i,di) (see Definition 6.1.1). It follows from the presen- C G = C G tation of a fundamental group that Π1 ( ,Γ) Π1 ( ,Γ ) (alternatively, one can easily check this using the universal property). This means that the described operation does not change the fundamental group; so by repeating this operation we arrive at a graph of pro-C groups (G0,Γ0) that satisfies the following property: whenever an edge e ∈ E(Γ0) has distinct vertices, one has that ∂i(G0(e)) = G0(di(e)),fori = 0, 1 C G = C G C and Π1 ( ,Γ) Π1 ( 0,Γ0). A graph of pro- groups satisfying this property is called reduced. When dealing with graphs of pro-C groups over finite graphs, the above construction allows us to restrict our considerations to reduced graphs of pro-C groups. Note that graphs of pro-C groups associated with free pro-C products of nontrivial groups and those associated with pro-C HNN-extensions (see Example 6.2.3(b), (e)) are automatically reduced. 228 7 Subgroups of Fundamental Groups of Graphs of Groups

Proposition 7.2.1 Let (G,Γ) be a reduced injective graph of pro-C groups over a = C G ﬁnite connected graph Γ . Then the action of Π Π1 ( ,Γ)on the standard graph S = SC(G,Γ)is irreducible.

Proof Note ﬁrst that, according to Theorem 6.5.2, SC(G,Γ) is a C-tree. Suppose that there exists a proper Π-invariant C-subtree D in S. Then the quotient Π\D is a proper subgraph of Γ . Since Γ is ﬁnite and connected, there exists an edge e in Γ − Π\D with one of its vertices w ∈ Π\D.Letv be the other vertex of e. Consider the edge e = 1Π(e) in S; its projection in Γ is e. Then e has vertices w and v , whose projections in Γ are w and v, respectively. Clearly w ∈ D and e ∈/ D. Therefore v ∈/ D,for otherwise the C-tree S/D (see Lemma 2.4.7) would contain a loop corresponding to e . Hence v ∈ Γ − Π\D. In particular, e has two different vertices.

We claim that Πv = Πe . To see this ﬁrst note that there exists some u ∈ D such that Πv ≤ Πu (see Corollary 4.1.9); therefore, by Corollary 4.1.6, Πv stabilizes every edge of the chain [v ,u ]; next observe that e ∈[v ,u ] (otherwise, after collapsing D and [v ,u ] to points, we would get a loop, contradicting Lemma 2.4.7); hence Πv ≤ Πe , and thus Πv = Πe , as claimed. Now, since e is not a loop, there is a maximal subtree T of Γ containing e; choose a fundamental 0-section j : Γ −→ Γ˜ of Γ˜ −→ Γ lifting T , and let J be the corresponding 0-transversal. Let (ν, ν ) be the universal J -specialization of (G,Γ) in C Π = Π (G,Γ). Since χj (e) = 1, we deduce (see formula (6.8) in Sect. 6.3) that the 1 end points of e = 1Π(e) in S are v = 1Π(v) and w = 1Π(w). Then di(e ) = v , for i = 0or1.So∂i(G(e)) ≤ G(di(e)) = G(v).UsingEq.(6.1) in Sect. 6.1,wehave

Πe = G(e) = ν G(e) = ν∂i G(e) ≤ ν G(v) = Π(v)= Πv .

Since Πe = Πv , it follows that ν∂i(G(e)) = ν(G(v)). Since ν is injective on G(v), we deduce that ∂i(G(e)) = G(v), contradicting the assumption that (G,Γ) is reduced.

Theorem 7.2.2 Let K be a ﬁnite normal subgroup of the fundamental pro-C group = C G C G Π Π1 ( ,Γ) of a reduced injective graph of pro- groups ( ,Γ) over a ﬁnite connected graph Γ . Then K ≤ Π(e), for every e ∈ E(Γ ).

= C G C = C G Proof Consider the action of Π Π1 ( ,Γ) on the pro- tree S S ( ,Γ) (see Theorem 6.5.2). This action is irreducible according to Proposition 7.2.1.If e ∈ E(Γ ), then e = 1Π(e)is an edge of S. Therefore the result follows from Propo- sition 4.2.2 and the fact that the Π-stabilizer of e is Π(e).

Corollary 7.2.3 (a) Let

G = G1 H G2

be a proper amalgamated free pro-C product of pro-C groups, G1 = H = G2. Then every ﬁnite normal subgroup of G is contained in H . 7.2 Normal Subgroups 229

(b) Let G = HNN(H,A,f) be a proper pro-C HNN-extension, A = H = f(A). Then every ﬁnite normal subgroup of G is contained in A.

Corollary 7.2.4 Let G beapro-C group and assume that it can be expressed as a free pro-C product

G = Gx x∈X of closed subgroups Gx continuously indexed by a proﬁnite space X. Assume that there exist x,y ∈ X, x = y, such that Gx = 1 = Gy . Then every ﬁnite normal subgroup K of G must be trivial.

Proof Write X = U ∪. V , where U and V are clopen disjoint neighbourhoods of x and y in X, respectively. Then G = G(U) G(V ), with

G(U) = Gu = 1 = Gv = G(V ) u∈U v∈V (see Lemma 5.5.2 and Lemma 5.5.1(b)). Think of G as the amalgamated free pro-C product of G(U) and G(V ) with amalgamated subgroup H = 1. Represent G as a fundamental pro-C group of a graph of groups as in Example 6.2.3(d). This graph of groups is injective and reduced (see Proposition 5.1.6(c)). By Corollary 7.2.3, K ≤ H , and so K = 1.

Lemma 7.2.5 Assume that C is an extension-closed pseudovariety of ﬁnite groups. = C G G C Let Π Π1 ( ,Γ), where ( ,Γ)is a reduced injective graph of pro- groups over a ﬁnite connected graph Γ such that |Γ | > 1. ∼ (a) If Π = ZCˆ, then the graph of groups (G,Γ)has the form

1 • 1

(b) Assume C ∈ C and let Π = L C =∼ C C C be the dihedral pro-C group, ∼2 2 2 where L = ZCˆ and C is a group of order 2; then the graph of groups (G,Γ)has the form C2•, 1 •C2

Proof Let S = SC(G,Γ) be the standard graph associated with (G,Γ). Since |Γ | > 1, one has |S| > 1. By Proposition 7.2.1, the action of G on S is irreducible, and by Theorem 6.5.2 S is a C-tree. We claim that, under the assumptions in (a) or (b), the action of Π on S is in addition faithful. Let K ={g ∈ Π | gs = s,∀s ∈ S} be the kernel of this action; we must show that K = 1. Define H = Π in case (a), and ∼ H = L in case (b). Then H = ZCˆ has at most index 2 in Π. Since every finite normal subgroup of Π is trivial (see Corollary 7.2.3), it suffices to show that K ∩ H = 1. 230 7 Subgroups of Fundamental Groups of Graphs of Groups

Observe ﬁrst that in case (b) K ∩ L

S −→ H/(K ∩ H) \S C ∩ = C ∩ \ is a universal -covering with H/(K H) π1 ((H/(K H)) S). Since the graph (H/(K ∩ H))\S is ﬁnite, its fundamental pro-C group is free pro-C (see Proposi- ∼ tion 3.5.3(b)), and since H/(K ∩ H)= 1 and H = ZCˆ,wemusthave ∼ H/(K ∩ H)= ZCˆ.

Therefore, the epimorphism H −→ H/(K ∩ H) is an isomorphism, since ZCˆ has the Hopfian property (see Sect. 1.3). Thus K ∩ H = 1, as desired. This proves the claim. In case (a), as pointed out above, Π = H acts freely on S. Hence, since the graph of groups (G,Γ) is reduced, the edges of the graph Γ must be loops, and Γ has a single vertex. It follows that =∼ =∼ C Π ZCˆ π1 (Γ ) and so Γ consists only of one loop; thus the graph of groups (G,Γ)is as indicated. In case (b) H has index 2 and acts freely on S. Therefore, the Π-stabilizers of any m ∈ S are of order at most 2. By Theorem 4.1.8 a finite subgroup of Π stabilizes a vertex of S; therefore Π is generated by its vertex stabilizers. Then, by Proposition 4.1.1, Γ = Π\S is a tree. So, since (G,Γ) is reduced, all edge groups =∼ C G of Π Π1 ( ,Γ)must be trivial. Therefore (see Example 6.2.3(a)), Π is a free pro-C product of finitely many copies of C2, one for each vertex of Γ . It follows from Corollaries 7.1.3 and 7.1.5 that the number of such copies is two. So Γ consists of one edge and two vertices, and the graph of groups (G,Γ)is as indicated.

To simplify the statement of the next theorem, we will say that a graph of pro-C groups (G,Γ) is of special type if it is one of the graphs of groups appearing in cases (a) or (b) of Lemma 7.2.5.

Theorem 7.2.6 Assume that C is an extension-closed pseudovariety of ﬁnite groups. Let (G,Γ)be a reduced injective graph of pro-C groups over a ﬁnite graph Γ which is not of special type. Let N be a normal subgroup of the fundamental pro-C = C G G ≤ ∈ group Π Π1 ( ,Γ)of ( ,Γ). Then either N Π(e) for every e E(Γ ), or else N contains an open subgroup H ≤o N having an epimorphism onto a nonabelian free pro-C group.

Proof Consider the action of Π on the standard graph S = SC(G,Γ). By Theo- rem 6.5.2, S is a C-tree and, by Proposition 7.2.1, Π acts irreducibly on S.Let 7.2 Normal Subgroups 231

K ={g ∈ G | gs = s,∀s ∈ S} be the kernel of this action. If N ≤ K, then N ≤ Π(e) for every e ∈ E(Γ ), and we are done. Otherwise, replacing Π by Π/K we may assume that Π acts faithfully on S and N is nontrivial. By Lemma 4.2.7, Π can be represented as an inverse limit = Π lim←− ΠW W∈W where the ΠW are quotient groups of Π, each acting faithfully and irreducibly on a C-tree DW with finite stabilizers of vertices and having a nontrivial open normal subgroup AW acting freely on DW . Moreover, according to statements (a) and (b) of that lemma, each DW is C-simply connected and ΠW \DW is finite, since Π\S = Γ is finite. It follows that AW \DW is finite. Therefore, by Proposition 3.5.3, AW = C \ C π1 (AW DW ) is a free pro- group. We claim that for some W ∈ W, AW is not procyclic. Indeed, suppose AW is procyclic, then ΠW does not have free nonabelian pro-p subgroups (since a free nonabelian pro-p group cannot have an open procyclic subgroup). Then, by Theo- ∼ rem 4.2.10, ΠW has one of the following forms: ΠW = Z ˆ Z ˆ with σ ∩ ρ =∅, ∼ σ ρ and GW acts freely on DW ; ΠW = Z ˆ Cn is a profinite Frobenius group; ∼ σ ΠW = Zσˆ C2 (2 ∈ σ ) is an infinite dihedral pro-σ group. However, only the first and third cases can occur since only in these cases can ΠW contain a free pro-C group of rank one as an open subgroup. But in the first case ΠW acts freely on DW and since ΠW \DW is finite, ΠW is free pro-C by Proposition 3.5.3; therefore in this ∼ case ΠW = ZCˆ. Hence if AW were procyclic for every W ∈ W, we would have that ΠW is either ZCˆ or Zσˆ C2 for all W ∈ W. It would follow that Π would be either free pro-C of rank one or infinite dihedral. In either case (G,Γ)would be of special type according to Lemma 7.2.5, a contradiction. This proves the claim. Hence we can choose W ∈ W so that AW is a free nonabelian pro-C group. Let NW be the image of N in ΠW . After replacing W , if necessary, by a convenient V ∈ W with V ≤ W , we may assume that NW is nontrivial. Then, by Proposi- tion 4.2.3(a), NW acts irreducibly on DW , and so, by Proposition 4.2.2, NW is infinite (observe that DW has more than one vertex since GW is infinite and its action on DW is irreducible). Therefore NW ∩AW is a nontrivial normal subgroup of the free nonabelian pro-C group AW . It follows from RZ, Theorems 8.7.1 and 8.6.5 or from RZ, Theorem 3.6.2 that NW ∩ AW contains an open free nonabelian pro-C group L. We can now let H be the preimage of L in N; since NW ∩ AW is open in NW ,wehavethatH is open in N.

Corollary 7.2.7 Assume that C is an extension-closed variety of ﬁnite groups such that the number of prime numbers p with Cp ∈ C is at least two: |π(C)|≥2. (a) Let Π be the pro-C fundamental group of a reduced injective graph of pro-C groups over a ﬁnite graph Γ which is not of special type. Then the Frattini subgroup Φ(Π) of Π is contained in every edge group Π(e) (e ∈ E(Γ )). 232 7 Subgroups of Fundamental Groups of Graphs of Groups

(b) Let {Gx | x ∈ X} be a family of pro-C subgroups of a pro-C group G continuously indexed by a proﬁnite space X, and assume that

G = Gx x∈X

(free pro-C product). Assume that there exist x1,x2 ∈ X, x1 = x2, such that = = { ∈ | = }={ } Gx1 1 Gx2 , and suppose in addition that if x X Gx 1 x1,x2 , = then eitherGx1 C2 or Gx2 C2. Then Φ(G) 1. (c) Φ(Z ˆ) = ∈ C pZ (we are identifying Z ˆ with ∈ C Z ). C p π( ) p ∼ C p π( ) p (d) Assume C2 ∈ C. Let G = K C = C2 C2 be the dihedral pro-C group, ∼ ∼ = ∼ K = ZCˆ and C = C2 acts on K by inversion. Then Φ(G) Φ(K) = p∈π(C) pZp.

Proof (a) Let p,q be different primes in π(C). The wreath product Cp Cq is generated by two elements and clearly it is not nilpotent since it does not have a unique q-Sylow subgroup; therefore, a nonabelian free pro-C group is not pronilpotent. Hence part (a) follows immediately from Theorem 7.2.6 and the fact that Φ(Π) is normal in Π and pronilpotent (cf. RZ, Corollary 2.8.4). To prove part (b), choose clopen neighbourhoods U1 and U2 in X of x1 and x2,re- . spectively, such that X = U1 ∪ U2. Write G as the free! pro-C product G =!G1 G2, where G and G are the free pro-C products G = G and G = G 1 2 1 x∈U1 x 2 x∈U2 x (see Lemma 5.5.2). Interpreting G = G1 G2 as the pro-C fundamental group of a graph of groups as in Example 6.2.3(b), the result follows from part (a). Part (c) is easy to check, and part (d) follows from (c).

7.3 The Kurosh Theorem for Free Pro-C Products

Here we deduce a pro-C version of the Kurosh subgroup theorem for open subgroups of free pro-C products of ﬁnitely many pro-C groups (see a different proof in RZ, Theorem 9.1.9). See also Theorem 9.6.2.

Theorem! 7.3.1 Let C be an extension-closed pseudovariety of finite groups. Let = n C C G i=1 Gi be a free pro- product of a finite number of pro- groups Gi . If H is an open subgroup of G, then n = ∩ −1 H H gi,τ Gigi,τ F i=1 τ∈H \G/Gi C ∩ −1 = isafreepro- product of groups H gi,τ Gigi,τ , where, for each i 1,...,n, gi,τ ranges over a system of representatives of the double cosets H \G/Gi , and F isafreepro-C group of finite rank rF , n rF = 1 − t + (t − ti), i=1 where t =[G : H ] and ti =|H\G/Gi|. 7.3 The Kurosh Theorem for Free Pro-C Products 233

Furthermore, one can arrange the choice of the giτ so that, for each i = 1,...,n, = ∩ there is some τi , such that giτi 1; that is, so that all the subgroups H Gi appear as free factors in the above decomposition of H .

C G C Proof Represent G as the fundamental group Π1 ( ,Γ) of a ﬁnite graph of pro- groups (G,Γ)as in Example 6.2.3(b)

G2 1 1 G 1 1 3 . 1 . . Gn We denote the vertices of Γ by ω,1,...,nand its edges by (ω, 1),...,(ω,n).The vertex groups of (G,Γ) in this case are G(ω) = 1, G(1) = G1,...,G(n) = Gn; and the edge groups G(ω, i) are all trivial. Denote by ν : G −→ G the canonical map. In this case ν is injective when restricted to each G(m) (m ∈ Γ). Hence we can make the identifications ν(G(i)) = G(i) = Gi (i = 1,...,n). Let C S = S (G,Γ)= . G/G(m) m∈Γ be the standard graph of the graph of groups (G,Γ). Since Γ is finite and H is open in G, the quotient graph Γ = H\S = . H\G/G(m) m∈Γ is finite. S

δ Γ = G\S Γ = H \S Let X ={1G(m) | m ∈ Γ }. According to Lemma 6.3.2, X is a subtree of S isomorphic to Γ . Hence X is isomorphic to the subtree Y = H1G(m) m ∈ Γ of Γ . Therefore, by Proposition 3.4.5 and Lemma 3.4.3, there exists a fundamental 0-section δ : Γ −→ S such that δ(H1G(m)) = 1G(m), for every m ∈ Γ . By The- C = H H = ∈ orem 6.6.1, H Π1 ( ,Γ ), where (m ) Hδ(m ),form Γ . Observe that in this case the edge groups of (H,Γ ) are trivial; so (see Example 6.2.3(a))

= C H = H H Π1 ,Γ (v) π1 Γ . v∈V(Γ ) 234 7 Subgroups of Fundamental Groups of Graphs of Groups

Fix m ∈ Γ ; then δ H\G/G(m) = gm,τ G(m) τ ∈ Im , and {gm,τ | τ ∈ Im} is a system of representatives of the double cosets H \G/G(m); furthermore, by our choice of δ, one has that

1 ∈{gm,τ | τ ∈ Im}. If m ∈ H \G/G(m) ⊆ Γ , then

δ m = gG(m), where g ∈{gm,τ | τ ∈ Im}, and

H m = Hδ(m ) = HgG(m) = H ∩ GgG(m) = H ∩ gG(m)g−1.

Since G(ω) = 1 and we are identifying G(i) with Gi when i = 1,...,n,wehave n = ∩ −1 H H gi,τ Gigi,τ F, i=1 τ∈Ii where F = πC(Γ ). Then F is a free pro-C group by Proposition 3.5.3(b), whose 1 rank is rF =|Γ |−|T |, where T isamaximaltreeofΓ . To compute this number, put t =[G : H ] and ti =|H\G/Gi|. Then n

|Γ |=(n + 1)t + ti. i=1

Since T is a ﬁnite tree, |T |= 2|V(T)|−1 = 2|V(Γ )|−1 (cf. Serre 1980, Propo- | |= + n − sition I.12); so T 2(t i=1 ti) 1. Therefore, n = − + − rF 1 t (t ti). i=1 Next we extend the above theorem to the restricted free pro-C product of a (possibly inﬁnite) collection of pro-C groups {Gs | s ∈ S} (see Example 5.1.9).

Corollary 7.3.2 Let H be an open subgroup of a restricted free pro-C product r G = Gs s∈S of a collection {Gs | s ∈ S} of pro-C groups. Then, for each s ∈ S, there exists a set Ds of representatives of the double cosets H\G/Gs such that the family of inclusions −1 H ∩ uGsu → H u ∈ Ds,s∈ S 7.3 The Kurosh Theorem for Free Pro-C Products 235 converges to 1, and H is the restricted free pro-C product & ' r −1 H = H ∩ uGsu F, s∈S,u∈Ds where F is a free pro-C group of ﬁnite rank. Furthermore, we may assume that for each s ∈ S, one has 1 ∈ Ds , so that H ∩ Gs appears as a free factor in the above decomposition of H .

Proof The idea is that the proof can be reduced to the case of a finite number of groups Gs . Consider the core −1 HG = gHg g∈G of H in G. Since H is open, so is HG. Hence, by definition of the restricted free pro- C product, there exists a finite subset B of S such that Gs ≤ HG for all s ∈ S − B.

Let G!be the closed subgroup of G generated by the groups {Gs | s ∈ S − B}; then = r G s∈S−B Gs . Therefore ( )

G = Gs G s∈B is a free pro-C product of finitely many factors, and one easily checks that it suffices to prove the theorem for this product: indeed, observe first that for all s ∈ S − B, HG ≥ Gs and since HG G, one has HuGs = Hu= HuG (u ∈ G), i.e., H \G/G = H\G = H\G/Gs ; on the other hand,

−1 −1 −1 −1 H ∩ uG u = uG u = uGsu = H ∩ uGsu s∈S−B s∈S−B (the restricted free pro-C product). Hence one may assume that S is a ﬁnite indexing set, and so the result follows from Theorem 7.3.1. Chapter 8 Minimal Subtrees

Throughout this chapter C is assumed to be an extension-closed pseudovariety of ﬁnite groups.

We are interested in a set-up of the following type. There is an abstract group R which is densely embedded in a pro-C group R˜; there is an abstract tree T which is densely embedded in a C-tree T˜ ; the group R˜ acts continuously on T˜ in such a way that R acts on T (i.e., T is R-invariant). Eventually we want to obtain some structural information about R out of the study of these actions and the relationship between T and T˜ . For this to have some hope of being meaningful and fruitful one obviously needs to have very close connections among all these actors, R, R˜, T and T˜ ; moreover those connections should have a bearing on the properties of R that one wants to investigate. To be concrete, let us consider the following example: C is the class of all finite groups; R is the fundamental group of a graph of finite groups (R,) over a finite connected graph ; R˜ = Rˆ, the profinite completion of R; T is the universal covering graph (an abstract tree) of (R,); and T˜ is the standard profinite graph (a profinite tree) of (R,); finally the property we want to investigate is the following: given x,y ∈ R, if one knows that the images of x and y in every finite quotient group of R are conjugate, is it true that x and y are conjugate in R? As we shall show later (see Theorem 14.1.4) the above set-up will provide a method to answer this question positively. In this chapter we consider a basic tool to study the connections among R, R˜, T and T˜ in some specific cases. Let H be a finitely generated subgroup of R and denote by H¯ its closure in R˜. In Proposition 2.4.12 we proved the existence of a unique minimal R˜-invariant subtree M˜ of R˜, under some conditions. Here we are interested in cases when T also admits a unique minimal H -invariant subtree M, and we investigate the relationship between M and M˜ in those specific cases.

© Springer International Publishing AG 2017 237 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_8 238 8 Minimal Subtrees

8.1 Minimal Subtrees: The Abstract Case

Lemma 8.1.1 Let R be an abstract group which is residually C. Assume that R acts freely on an abstract tree T . Endow R with its pro-C topology. Let K be a closed subgroup of R in this topology and let be a ﬁnite subgraph of the quotient graph K\T . Then there exists an open subgroup V of R containing K such that the natural map of graphs

τV : K\T −→ V \T is injective on .

Proof Consider the map of graphs τR : K\T −→ R\T . Since is finite, it is a finite union of intersections as follows: m = ∩ −1 τR (xt ) , t=1 for some xt ∈ R\T and some natural number m. We claim that for each t = : 1,...,m, there exists some open subgroup Vt of R containing K such that τVt \ −→ \ ∩ −1 K T Vt T is injective on τR (xt ). Since R acts freely on T ,theset −1 \ ≤ ≤ τR (xt ) may be identified with K R; moreover, if K U R, the restriction of : \ −→ \ −1 τU K T U T to τR (xt ) may be identified with the canonical surjection τU : K\R −→ U\R. Now, since K is a closed subgroup of R, it is the intersection of all the open subgroups of R containing K (cf. RZ, Lemma 3.1.2): K = U.

K≤U≤oR ∩ −1 \ Since τR (xt ) can be thought of as a ﬁnite subset of K R, the existence of the = m required Vt follows, proving the claim. Deﬁne V to be V t=1 Vt . Then clearly τV is injective on .

Lemma 8.1.2 Let R be an inﬁnite abstract group that acts on an abstract tree T in such a way that the stabilizer of each edge is ﬁnite. Then T cannot contain disjoint R-subtrees. Consequently, if T has a minimal R-invariant subtree, it is unique.

˜ Proof Let T1 and T2 be disjoint R-subtrees of T . Consider the graph T obtained ˜ from T by collapsing T1 and T2 to distinct vertices v1 and v2, respectively. Then T is a tree (cf. Serre 1980, Corollary 2 to Proposition I.13) on which R acts with finite edge stabilizers and the vertices v1 and v2 are fixed by R.Letp be a path joining ˜ v1 and v2 in T . Since R is infinite and the stabilizers of edges are finite, one can ˜ choose r ∈ R, so that p and rp are different paths joining v1 and v2 in T . One gets a contradiction to the fact that T˜ is a tree by observing that the underlying graph of p ∪ rp contains a circuit. 8.1 Minimal Subtrees: The Abstract Case 239

Next we consider a special type of minimal subtree associated to certain elements of the abstract fundamental group of a graph of groups. A subtree L of an abstract tree is called a line if its underlying geometric graph (obtained after disregarding the orientation of the edges) is doubly infinite of the form ···• • • •··· If v and w are vertices of an abstract tree T ,thedistance d(v,w) between v and w is the length of the unique reduced path joining v and w.Ifv is a vertex of T and T is a subtree of T ,thedistance d(v,T) between v and T is the infimum of all d(v,w) for w ∈ V(T ). Note that there is a unique v ∈ V(T ) with d(v,T ) = d(v,v ). Next we recall a result of J. Tits that we state in a manner convenient to us. Let G ∈ abs = abs G ( ,) be a graph of groups. We say that an element b Π Π1 ( ,) is hyperbolic if it does not fix any vertex of the tree Sabs = Sabs(G,); in other words, b does not belong to a conjugate of a subgroup Πabs(v) in Π abs, for any v ∈ V().

Proposition 8.1.3 Let b ∈ Π abs be a hyperbolic element. Put abs abs m = inf d(v,bv) v ∈ V S and Lb = v ∈ V S d(v,bv) = m . Then

(a) Lb is the set of vertices of a b-invariant line (the ‘Tits line corresponding to b’) which we denote again by Lb. The action of b on Lb determines a translation of amplitude m on the underlying geometric graph of Lb; abs (b) Lb is contained in any b-invariant subtree of S ; in fact Lb is the unique minimal b-invariant subtree of Sabs; (c) if v ∈ Lb, then Lb = b[v,bv]; abs (d) if w ∈ V(S ) is at a distance t of Lb, then d(w,bw)= m + 2t.

Proof Since b is hyperbolic, m ≥ 1. Let v beavertexofLb.Letv0 = v,e1,v1,..., vm−1,em,vm = bv be the sequence of vertices vi and edges ei of the chain c = [v,bv], where the edge ei has vertices {vi−1,vi}. We claim that the sequence obtained by concatenating c =[v,bv] and bc =[bv,b2v]: 2 v0 = v,e1,v1,...,vm−1,em,vm = bv,be1,bv1,...bvm−1,bem,bvm = b v is the sequence of vertices and edges of the chain [v,b2v]. Indeed, if this were not the case, one would have that for some i>0, bvi = vj , where j1, the distance between v1 and bv1 would be m − 2, contradicting the minimality of m. This proves the claim. It follows immediately by induction that the concatenation of the chains bn[v,gv] (n ∈ Z) form a line L, and b acts on L as a translation of amplitude m. abs Next we show that L = Lb. To see this, let w be a vertex of S not in L, and let v ∈ L be such that d(w,v ) = t is minimal; note that v is unique. Then, since Sabs is a tree, the length of [w,bw]=[w,v ]∪[v ,bv ]∪[bv ,bw] is d(w,bw) = 2t + m. Hence d(w,bw) = m if and only if w ∈ L; therefore L = Lb. One easily checks now all parts of the proposition. 240 8 Minimal Subtrees

Remark 8.1.4 If T is an abstract tree and ϕ : T → T is an automorphism which does not ﬁx any vertex, then one can prove a result analogous to the proposition above, essentially along the same ideas: there is a unique line in T , usually called the axis of ϕ, on which ϕ induces a translation of amplitude

m = inf d x,ϕ(x) . x∈V(T)

Corollary 8.1.5 Let b ∈ Π abs be a hyperbolic element. Assume that L is a b- abs invariant line in S . Then L = Lb.

Lemma 8.1.6 Let b ∈ Π abs be a hyperbolic element and assume that c = xbx−1, abs where x ∈ Π . Then c is also hyperbolic, Lc = xLb and the amplitudes of the translations determined by b and c are the same.

Proof Choose v ∈ V(Lb);sayd(v,bv) = m. Note that c acts on the line xLb as a translation of amplitude m. By Corollary 8.1.5, Lc = xLb.

Example 8.1.7 (Tits lines in the standard tree of a free product of groups) Let G = G1 ∗ G2 be the free product of groups G1 and G2; in the language used above, G is the fundamental group of the graph of groups

1 (G,)= G1 G2 . Consider the universal covering graph Sabs of this graph of groups. ...

g2g1G2 g2g1

g1G2 g2G1 g1 g2

1 1G1 1G2

g2G1

Let b ∈ G = G1 ∗ G2 be an element of G which is not in a conjugate of an element of either G1 or G2, i.e., b is a hyperbolic element in the sense mentioned above. Write b = xcx−1, where x,c ∈ G and where = ··· c gi1 gi2 git , ≥ = ∈ ∪ ∈{ } = = t 2, 1 gij G1 G2, ij 1, 2 , ij ij+1, i1 it . Note that t is uniquely determined by b. Observe that c is also hyperbolic. For deﬁniteness, say that i1 = 2 8.2 Minimal Subtrees: Abstract vs Proﬁnite Trees 241

abs and put v = 1G2. Then (see Corollary 8.1.5) the Tits line in S corresponding to c is

Lc = c[v,cv]. abs According to Lemma 8.1.6, the Tits line in S corresponding to b is Lb = xLc.In other words,

Lb = b[xv,bxv].

The amplitude of the translation on Lb induced by the action of b is t.

8.2 Minimal Subtrees: Abstract vs Proﬁnite Trees

We begin with a general result that will serve as the basis for cases that we treat later. Consider now the following situation: Let H be an abstract group which is embedded as a dense subgroup in a nonfinite pro-C group H˜ . Assume that T abs is an abstract tree which is embedded as a dense subgraph of a C-tree T . We assume further that H˜ acts continuously on the C-tree T in such a way that T abs is H - invariant and such that H\T abs is a finite graph, and suppose that if T abs has an H -invariant minimal subtree, then it is unique (this happens if, for example, the H -stabilizer of each edge is finite, according to Lemma 8.1.2).

Lemma 8.2.1 Assume in addition that the natural epimorphism of graphs H\T abs −→ H˜ \T is an isomorphism. Then there exists a unique minimal H -invariant subtree Dabs of T abs and its closure D = Dabs in T is the unique minimal H˜ -invariant C-subtree of T ; moreover, Dabs = T abs ∩ D and H\Dabs = H˜ \D is ﬁnite.

Proof We shall identify H\T abs with H˜ \T . Consider the commutative diagram T abs T

ηabs η H\T abs = H˜ \T where ηabs and η are the canonical quotient maps of graphs. We observe that this means that if x,y ∈ T abs and x ∈ Hy˜ , then x ∈ Hy.TheH -invariant subgraphs of T abs have the form (ηabs)−1(X), for some subgraph X of H \T abs, and, similarly, H˜ -invariant proﬁnite subgraphs of T have the form η−1(X). Since H \T abs is a ﬁnite graph, we deduce that T abs has a minimal H -invariant subgraph, which is unique by assumption. We remark that if T is an H˜ -invariant C-subtree of T , then T abs ∩ T is obviously H -invariant, and T abs ∩ T = ∅ (for (ηabs)−1(η(T )) ⊆ T abs ∩ T ), so that T abs ∩ T isasubtreeofT abs (for if v and w are vertices of T abs ∩ T , then [v,w]⊆T and also [v,w]⊆T abs). Furthermore, T abs ∩ T = T : indeed, let 242 8 Minimal Subtrees

Σ be a 0-transversal of η(T ) in T abs with respect to the H -action (i.e., an H - transversal with d0(m) ∈ Σ , for each m ∈ Σ ); then by our hypothesis, Σ is also a 0-transversal of η(T ) in T with respect to the H˜ -action; hence T abs ∩ T = HΣ and T = HΣ˜ , and in particular T abs ∩ T = T . Let D be the unique H˜ -invariant C-subtree of T (see Proposition 2.4.12). Put Dabs = T abs ∩ D. We claim that Dabs is the unique minimal H -invariant subtree of T abs. Indeed, if abs were a proper H -invariant subtree of Dabs, we would be able to choose 0-transversals Σ and Σ of ηabs(abs) and ηabs(Dabs), respectively, under the action of H , so that Σ ⊂ Σ . Then by the above remark and the above observation, HΣ˜ would be a proper H˜ -invariant subtree of D, contradicting the minimality of D. This, together with the assumption that minimal H -invariant subtrees are unique, proves the claim. Finally, since H \Dabs ⊆ H\T abs and H˜ \D ⊆ H˜ \T , we deduce that H \Dabs = H˜ \D and this graph is ﬁnite.

Trees Associated with Virtually Free Groups

Let P be a property of groups. Recall that a group R is said to be virtually P if R contains a subgroup of finite index with property P. Thus we speak of a virtually free group, or a virtually torsion-free group, or a virtually abelian group, etc. Let R be a finitely generated abstract free-by-C group, i.e., R contains a normal subgroup Φ which is a free abstract group of finite rank and R/Φ ∈ C. Note that R is in particular virtually free. We describe next the construction of an abstract abs tree S and a C-tree S associated with R and with the pro-C completion RCˆ of R, respectively, so that Sabs is a dense subgraph of S. We begin with a preliminary result.

Lemma 8.2.2 Let R be an abstract free-by-C group. Then the pro-C topology of R is Hausdorff, i.e., R is residually C. In particular, R ≤ RCˆ.

Proof Let Φ be a normal free subgroup of R such that R/Φ ∈ C.Let1= x ∈ R.We need to prove the existence of a normal subgroup U of R such that R/U ∈ C and x/∈ U.Ifx/∈ Φ, choose U = Φ. Assume x ∈ Φ. The pro-C topology of Φ coincides with the topology induced by the pro-C topology of R (cf. RZ, Lemma 4.1.4(a)); on the other hand, the pro-C topology of a free abstract group is Hausdorff (cf. RZ, Proposition 3.3.15); hence there exists a normal subgroup U of R with R/U ∈ C, U ≤ Φ and x/∈ U.

Proposition 8.2.3 Let R be a finitely generated abstract free-by-C group. Then the following assertions hold. abs = abs G (a) R is the abstract fundamental group Π Π1 ( ,) of a graph of groups (G,)over a finite graph such that each G(m) (m ∈ ) is a finite group in C. 8.2 Minimal Subtrees: Abstract vs Profinite Trees 243

C = C G G C (b) The pro- fundamental group Π Π1 ( ,)of ( ,)is the pro- completion RCˆ of R. (c) In this case the canonical homomorphisms νm : G(m) −→ Π(m∈ ) are em- beddings. (d) In this case, the abstract tree Sabs = Sabs(R,)is canonically embedded in the C-tree S = SC(R,)by means of the map

gΠabs(m) → gΠ(m) m ∈ ,g ∈ R = Π abs .

Proof To fix the notation, let Φ be a normal free subgroup of R such that R/Φ ∈ C. (a) According to a result of Karrass, Pietrowski and Solitar (cf. Karrass, Pietrowski and Solitar 1973, Theorem 1), R is the abstract fundamental group abs = abs R R Π Π1 ( ,) of a graph of groups ( ,) over a finite graph such that each R(m) (m ∈ ) is a finite group. The isomorphic image Π abs(m) of R(m) (m ∈ ) is a subgroup of R = Π abs; on the other hand, a finite subgroup of R is isomorphic to a subgroup of R/Φ, and so it is in C. Parts (b), (c) and (d) follow from Proposition 6.5.5.

Proposition 8.2.4 We continue with the hypotheses and notation of Proposi- tion 8.2.3. Let H = h1,...,hr be an infinite subgroup of the free-by-C group R = Π abs which is finitely generated and closed in the pro-C topology of R, and let ¯ H be its closure in the pro-C group RCˆ = Π. Then the following assertions hold. (a) Sabs has a unique minimal H -invariant subtree Dabs, and its closure D in S is the unique minimal H¯ -invariant C-subtree of S. (b) Sabs ∩ D = Dabs. (c) H \Dabs = H¯ \D is finite. (d) If β ∈ H¯ and βw ∈ Dabs for some w ∈ Dabs, then β ∈ H . (e) If β ∈ H¯ − H , then βDabs ∩ Sabs =∅. ¯ (f) Let {βλ | λ ∈ Λ} be a complete set of representatives of the left cosets of H in H (a transversal). Then . abs D = βλD . λ∈Λ abs In other words, the abstract graphs βλD are the distinct abstract connected components of D considered as an abstract graph; in particular, Dabs is an abstract connected component of D as an abstract graph.

Proof Let Φ be a normal free subgroup of R such that R/Φ ∈ C. Choose a vertex abs abs v0 of , and denote by v˜0 the vertex v˜0 = 1Π (v0) = 1Π(v0) in S ⊆ S. Define a subgraph T abs of Sabs as follows r abs T = H[˜v0,hiv˜0]. i=1 = r [˜ ˜ ] abs = Put L i=1 v0,hiv0 ; this is obviously a finite connected graph, and T HL. abs Since L ∩ hiL = ∅ (i = 1,...,r),wehavethatT is a connected subgraph of the 244 8 Minimal Subtrees tree Sabs (see Lemma 2.2.4), and so T abs is a tree; clearly it is H -invariant. Hence its closure in S r abs ¯ T = T = H[˜v0,hiv˜0] i=1 is a C-subtree of S (see Lemma 2.1.7 and Proposition 2.4.3(b)); clearly it is H¯ - invariant. Since H is infinite and each G(m) is finite, one deduces from Lemma 8.1.2 that if Sabs has a minimal H -invariant subtree, it coincides with the unique minimal H -invariant subtree of T abs. Therefore parts (a), (b) and (c) will follow from Lemma 8.2.1 after we show that the epimorphism of graphs H \T abs −→ H¯ \T is in fact an isomorphism. To see this we distinguish two cases.

Case 1. Assume that H ≤ Φ. Since the R-stabilizers of the elements of Sabs are finite groups, we have that Φ acts freely on Sabs. By Lemma 8.1.1, there exist an open subgroup V of Φ (and so of Π abs) containing H such that the map of graphs H\T abs −→ H\Sabs −→ V \Sabs is injective. Next observe that for every m ∈ , we have the following equality of double cosets V \Π abs/Π abs(m) = V¯ \Π/Π(m) because Π abs(m) = Π(m), V has finite index in Π abs and Π is the pro-C completion of Π abs; hence, one deduces that V \Sabs = V¯ \S from the definition of Sabs and S. Therefore, from the commutative diagram H\T abs H\Sabs V \Sabs

H¯ \T H¯ \S V¯ \S we deduce that H \T abs = H¯ \T , proving the result in this case.

Case 2. General case. Define K = Φ ∩ H . Note that K is closed in Φ and that K\T abs is finite (because K has finite index in H ). So Lemma 8.1.1 can be used. Mimicking the argument in Case 1 one shows that K\T abs = K¯ \T . What this says is that if t,t ∈ T abs, and ¯ ¯ Kt = Kt , then Kt = Kt . . Now since K has finite index in H , we have finite unions H = Kxi and ¯ . ¯ abs ¯ ¯ H = Kxi (some xi ∈ H ). Let t,t ∈ T , and assume that Ht =Ht . We want ¯ ¯ to show that then Ht = Ht . By hypothesis we have Kxit = Kxit .Sofor

¯ = ¯ ¯ = ¯ each i, there are some i and i such that Kxit Kxi t and Kxit Kxi t; hence

Kxit = Kxi t and Kxit = Kxi t. Therefore, Kxit = Kxit , i.e., Ht = Ht . This completes the proof of (a), (b) and (c). 8.3 Graphs of Residually Finite Groups and the Tits Line 245

(d) Say w = aΠabs(m) = aΠ(m) (a ∈ Π abs,m∈ ). Assume that βw ∈ Sabs. Since βw and w have the same image in H¯ \D, we deduce from part (c) that there exists some h ∈ H with hw = βw.Soh−1βw = w. Then h−1β is in the Π-stabilizer of w, i.e., h−1β ∈ aΠ(m)a−1 = aΠabs(m)a−1 ≤ Π abs. Therefore β ∈ Π abs ∩ H¯ = H , because H is closed. (e) Using part (b) we have βDabs ∩ Sabs ⊆ D ∩ Sabs = Dabs. Hence the result follows from (d). (f) Clearly ¯ abs . abs abs D = HD = βλH D = βλD . λ∈Λ λ∈Λ This last union is disjoint by (d).

8.3 Graphs of Residually Finite Groups and the Tits Line

The aim of this section is to prove a partial generalization of Proposition 8.2.4.On the one hand, we will allow the vertex groups of the graph of groups (G,) to be infinite, but on the other hand, we only consider the class C of all finite groups and we restrict the type of groups H for which we seek minimal H -invariant subtrees, namely we assume that H is cyclic of a special type. The general set-up for this section is the following. Let be a finite connected graph and let (G,) be a graph of abstract groups over . Assume that its funda- abs = abs G abs mental group Π Π1 ( ,) is residually finite and each Π (m) is closed in the profinite topology of Πabs. As in Sect. 6.5,let(G¯,) be the graph of profinite groups over , where G¯(m) is the completion of G(m) with respect to the topology abs ¯ induced from the profinite topology of Π .LetΠ = Π1(G,)be the corresponding profinite fundamental group. Recall that (G¯,)is an injective graph of profinite groups. Let Sabs = Sabs(G,) (respectively, S = S(G¯,)) be the standard tree (respectively, standard profinite tree) of this graph of groups. Then Sabs is densely embedded as a subgraph of S.

Proposition 8.3.1 We continue with the above set-up (C is the class of all ﬁnite abs abs groups). Let b ∈ Π be a hyperbolic element of Π and let Lb be the corresponding Tits line. Then the following assertions hold. n n (a) b \Lb = b \Lb, for all natural numbers n = 1, 2,.... n (b) Lb is the unique minimal b -invariant proﬁnite subtree of S and Lb ∩ abs S = Lb. (c) If β ∈ b and βw ∈ Lb for some w ∈ Lb, then β ∈ b. abs (d) If β ∈ b− b, then βLb ∩ S =∅. (e) Let {βλ | λ ∈ Λ} be a complete set of representatives of the cosets of b in b (a transversal). Then . Lb = βλLb. λ∈Λ 246 8 Minimal Subtrees

In other words, the abstract graphs βλLb are the distinct connected components of Lb considered as an abstract graph; in particular, Lb is its own connected component in Lb as an abstract graph. abs (f) Let N ={x ∈ Π | xLb = Lb}. Then N is closed in the proﬁnite topology of Π abs.

Before embarking on the proof of this proposition, we need two auxiliary results. The strategy in the proof of Proposition 8.3.1 above is to find a way to use Proposition 8.2.4; one cannot use it directly because the latter assumes that the fundamental group Π abs is free-by-finite; however we show in our second auxiliary result (Lemma 8.3.3) that, under the hypotheses of the above proposition, Π can be expressed as an inverse limit of profinite fundamental groups which are the profinite completions of abstract free-by-finite groups. In fact, these two auxiliary results are proved in more generality than needed for the purpose of handling Proposition 8.3.1: we prove them for pro-C groups, where C is extension-closed, and instead of the profinite topology, we consider the pro-C topology, etc.

Lemma 8.3.2 Assume that C is extension-closed. We continue with the set-up of this section. Let b ∈ Π abs be hyperbolic. Then b acts freely on the standard C-tree C ¯ ¯ ∼ S = S (G,)of the graph of pro-C groups (G,). Furthermore, b = ZCˆ.

Proof Let Lb be the Tits line corresponding to b, and choose a vertex v in Lb.By Proposition 8.1.3, Lb = b[v,bv], and therefore Lb = b[v,bv]. Since Lb is an ab- abs stract subtree of the tree S , we deduce that Lb is a connected proﬁnite subgraph of the C-tree S (see Lemma 2.1.7(a)). Since C is extension-closed, S is C-simply connected (and so a C-tree; see Theorem 6.3.5 and Corollary 6.3.6); therefore so is Lb (see Proposition 3.7.3(d)). According to Proposition 8.1.3, b acts freely on Lb.

We claim that b acts freely on Lb.Letb ∈ b and assume that b fixes some vertex b w of Lb (b ∈ b,w ∈ V([v,bv])). Then b b w = b b w = b w;so b w = w; hence b (bw) = bw. Therefore unless all the elements of [w,bw] are fixed by b , [w,bw]∪b [w,bw] would contain a circuit; since the latter is impos- sible because S is a C-tree, we deduce that b fixes all elements of [w,bw] and so of Lb = b[w,bw]. In other words, we have shown that if b fixes an element of Lb, then it acts trivially on Lb.LetK consist of all elements of b that act trivially on Lb; observe that K is a closed subgroup of b. Then b/K acts freely on Lb. Since, as pointed out above, Lb is C-simply connected and the quotient graph ( b/K)\Lb is finite, it follows from Corollary 6.6.2 that b/K is a free pro-C ∼ group. Since b/K is procyclic, we deduce that b/K = ZCˆ. Therefore there are epimorphisms ∼ b−→ b/K = ZCˆ −→ b. The composition of these epimorphisms must be an isomorphism (see Sect. 1.3), and hence so is b→ b/K. Thus K = 1. This proves the claim. The result then follows from Corollary 4.1.7(c). 8.3 Graphs of Residually Finite Groups and the Tits Line 247

Lemma 8.3.3 Assume that C is extension-closed. We continue with the set-up of this section. Let U be the collection of all open (in the pro-C topology) normal sub- abs = abs G groups of Π Π1 ( ,). Then there is an inverse system of graphs of groups (GU ,)over satisfying the following conditions. ¯ (a) Each G (m) is a group in C (m ∈ ) and (G,)= lim ∈U (G ,). U ←− U U ∈ U = C G C (b) For each U , let ΠU Π1 ( U ,) be the pro- fundamental group of the C graph of groups (GU ,), and let SU = S (GU ,) be the corresponding standard C-tree. Then = = Π lim←− ΠU and S lim←− SU , U∈U U∈U where Π is the fundamental pro-C group of (G¯,), and where S is the corresponding standard C-tree. ∈ U abs = abs G (c) For each U , the abstract fundamental group ΠU Π1 ( U ,) of the graph of groups (GU ,)contains an open (in its pro-C topology) free subgroup, C abs abs so that ΠU is the pro- completion of ΠU , and the standard abstract tree SU corresponding to (GU ,)is densely embedded in SU . (d) The canonical projections ϕU : Π −→ ΠU and ψU : S −→ SU are compatible with the actions of Π on S and of ΠU on SU , i.e., ψU (ga) = ∈ ∈ abs = abs ϕU (g)ψU (a), (g Π,a S); furthermore, ϕU (Π ) ΠU and abs = abs ∈ U ψU (S ) SU , for each U . (e) Let H be a closed subgroup of Π that acts freely on S. Then there exists some ˜ ˜ U ∈ U such that for all V ∈ U with V ≤ U one has that ϕV (H ) acts freely on SV .

Proof For each U ∈ U, consider the graph of groups (GU ,)over with abs abs GU (m) = Π (m)/Π (m) ∩ U(m∈ ). The only parts that require an explicit proof are (c) and (e), for (a), (b) and (d) are immediate consequences of the deﬁnitions. (c) For U ∈ U, denote by U˜ the subgroup of Π abs generated by the U-stabilizers of the vertices of Sabs, i.e., U˜ = U ∩ gΠabs(v)g−1 v ∈ V(),g∈ Π abs . Clearly U˜ Πabs. Then U˜ \Sabs is a tree (this follows from Serre 1980, Corollary 1 to Theorem I.13: explicitly, see Exercise 2 on that page). Now, Π abs/U˜ acts on U˜ \Sabs and * Π abs/U˜ U˜ \Sabs = Π abs\Sabs = . Furthermore, the Π abs/U˜ -stabilizer of the vertex U˜ 1Π abs(m) of U˜ \Sabs is UΠ˜ abs(m)/U˜ =∼ Π abs(m)/U˜ ∩ Π abs(m) = Π abs(m)/U ∩ Π abs(m). Therefore, abs ˜ = abs G = abs abs = ˜ \ abs Π /U Π1 ( U ,) ΠU and SU U S 248 8 Minimal Subtrees

(cf. Serre 1980, Theorem I.13). Finally, observe that U/U˜ acts freely on the tree U˜ \Sabs, and hence it is free (cf. Serre 1980, Theorem I.4); obviously U/U˜ is open C abs abs ˜ ˜ =∼ abs ∈ C in the pro- topology of ΠU , since (Π /U)/(U/U) Π /U . (e) Let Y (respectively, YU , where U ∈ U) be the compact subspace of the points of S (respectively, SU )ﬁxedbyH (respectively, ϕU (H )); clearly = Y lim←− YU . U∈U Since H acts freely on S, Y =∅. By compactness, there exists some U˜ ∈ U such ˜ that YV =∅whenever V ∈ U and V ≤ U (see Sect. 1.1). That is, ϕV (H ) acts freely on SV for all such V .

Proof of Proposition 8.3.1 (a) Case 1. n = 1. According to Lemma 8.3.2, b acts freely on the profinite tree S = SC(G¯,), and so b has infinite order (see Theorem 4.1.8). We continue with the notation of Lemma 8.3.3.LetbU = ϕU (b) denote the image of b in ΠU . By Lemma 8.3.3(e), ˜ ˜ there exists some U ∈ U such that bV acts freely on SV for every V ≤ U; in partic- ∼ ular, we may assume that each bV has infinite order. We claim that bV = Z.Tosee abs this it suffices to show that the profinite topology of ΠV induces on bV its full profinite topology. Let FV be an open (in the profinite topology) free subgroup of abs abs ΠV (see Lemma 8.3.3(c)). Then ΠV induces on FV its full profinite topology (cf. RZ, Lemma 3.1.4). Now by a result of M. Hall, FV ∩ bV is a free factor of a subgroup of finite index in FV (cf. Lyndon and Schupp 1977, Chap. I, Theorem 3.10). abs Therefore the profinite topology of FV (and so of ΠV ) induces on bV its full profinite topology (cf. RZ, Lemma 3.1.4 and Corollary 3.1.6), proving the claim. One deduces that the natural epimorphism b−→ bV is an isomorphism. abs = abs abs = abs ∈ U By Lemma 8.3.3(d) ϕU (Π ) ΠU and ψU (S ) SU . For every U , abs ϕU (Lb) is connected and so is a subtree of the tree SU which is bU -invariant; ⊆ therefore, LbU ϕU (Lb). ∈ [ ] = Choose a vertex v V(Lb) of Lb. Since v,bv is finite and since S ←−lim SU , there exists some U0 ∈ U such that the restriction of ψU to [v,bv] is an injec- ˜ tion for every U ≤ U0. Choose U ≤ U0 ∩ U. Hence ψU sends Lb = b[v,bv] onto ψU (Lb) = bU [ψU (v), bU ψU (v)] bijectively and hence ψU (Lb) is min- ⊆ imal. Since LbU ϕU (Lb), we deduce that ψU sends the b -space Lb to the = bU -space LbU isomorphically; similarly ϕU (Lb) LbU . One knows that bU abs is closed in the profinite topology of ΠU (we prove this later in Theorem 11.2.2); \ = \ \ = hence we can use Proposition 8.2.4 to get bU LbU bU LbU ;so b Lb b\Lb.

Case 2. n>1. n Observe that Lbn = Lb. Since b is hyperbolic, the result in this case follows from Case 1. This completes the proof of (a). (b) This follows from part (a) (for n = 1) and Lemma 8.2.1. 8.3 Graphs of Residually Finite Groups and the Tits Line 249

(c) Say βw = v ∈ Lb. By part (a), there exists some b1 ∈ b with b1w = v. −1 −1 So β b1w = w. Then β b1 = 1 because b acts freely on S according to Lemma 8.3.2. Thus β = b1 ∈ b. abs abs (d) Using part (b) we have βLb ∩S ⊆ Lb ∩S = Lb. Hence the result follows from (c). (e) Clearly . Lb = bLb = βλ b Lb = βλLb. λ∈Λ λ∈Λ This last union is disjoint by (c). (f) Put M ={x ∈ Π | xLb = Lb}. By the compactness of Lb and the continuity of the action, we have that M is a closed subgroup of Π. Statement (f) would follow if we can prove that M ∩ Π abs = N. Clearly N ≤ M ∩ Π abs. Conversely, assume that abs abs x ∈ M ∩ Π ; we need to show that x ∈ N. Note that xLb ⊆ Lb ∩ S ; therefore, −1 abs −1 using part (b), xLb ⊆ Lb. Since also x ∈ M ∩ Π ,wehavex Lb ⊆ Lb. So, xLb = Lb, i.e., x ∈ N.

Note that we have proved Proposition 8.3.1 only when the class C consists of all finite groups and so the topology considered is the profinite topology. The proof that we have presented here does not appear to be valid for extension-closed classes C in general: the problem seems to appear at the end of the proof of Case 1 of part (a), where one needs the group bU to be closed; this is automatic for the profinite topology, but not for the pro-C topology in general.

Corollary 8.3.4 Let Lb be as in Proposition 8.3.1. Then the profinite tree Lb does not contain proper infinite profinite subtrees.

Proof Note that = n \ Lb lim←− b Lb. n∈N By Proposition 8.3.1(a), bn\L = bn\L so that b b = n \ Lb lim←− b Lb. n∈N Now we proceed as in Example 2.1.13.Letm be the length of [v,bv], where v is n avertexofLb. Observe that Γn = b \Lb is a circuit of length nm.Let be a proper proﬁnite subtree of Lb. Then there exists some n0 such that the image n0 of in Γn is proper. Since n is connected, it is a chain which does not coincide 0 0 | : → with Γn0 .Letn0 n , and denote by ρn n0 Γn Γn0 the natural projection. Then −1 ρ ( ) is the disjoint union of n /n chains, each of them isomorphic to n n n0 0 0 0 under the map ρn n0 . Hence the image n of in Γn must be one of those chains. It follows that is isomorphic to n0 , and therefore it is ﬁnite.

Proposition 8.3.5 Let (G,) be a graph of abstract groups over a ﬁnite graph and assume that 250 8 Minimal Subtrees

= abs G (i) R Π1 ( ,)is residually finite, (ii) the profinite topology of R induces on each Π abs(m) (m ∈ ) its own full profinite topology, and (iii) each Π abs(m) (m ∈ ) is closed in the profinite topology of R. Let a,b ∈ R, and assume that a is hyperbolic (i.e., a does not fix any vertex of the tree Sabs = Sabs(G,)). Assume that b is conjugate to a in Rˆ: a = bγ = γ −1bγ , where γ ∈ Rˆ. Then

(a) b is also hyperbolic and Lb = γ La; and abs ˆ (b) if e is an edge of La, then there exist x ∈ R and βe ∈ Π(e)= Π (e) (= (R)e, the Rˆ-stabilizer of e) such that −1 = −1 x bx βe aβe.

Proof Continuing with the notation of this section, observe that under our assumptions G¯(m) = G(m) =∼ Π(m)= Π abs(m) (m ∈ ). (a) By Lemma 8.3.2 a acts freely on S. Hence b = γaγ−1 acts freely on S = γS, and in particular on Sabs. Therefore b is hyperbolic. −1 Clearly b=γ aγ . Since La is the unique minimal a-invariant proﬁnite subtree of S (see Proposition 8.3.1(b)), γ La is the unique minimal b-invariant proﬁnite subtree of S; therefore

Lb = γ La.

(b) From (a) one has γe∈ Lb.Letv be a vertex of Lb; then Lb = b[v,bv], ˜ ˜ so that Lb = b[v,bv]. Therefore, there exists a b ∈ b such that bγ e ∈[v,bv]⊆ Sabs. Since bγ˜ e and e have the same image in = Rˆ\S = R\Sabs, there exists some x ∈ R with bγ˜ e = xe. ˜ = −1 ∈ ˆ So bγ xβe , where βe Π(e) (note that Π(e) coincides with the e-stabilizer Re under the action of Rˆ). Hence, taking into account that b and b˜ commute, = −1 = −1 ˜−1 ˜ = −1 −1; a γ bγ γ b bbγ βex bxβe −1 = −1 thus, x bx βe aβe, as desired.

8.4 Graph of a Free Product of Groups and the Tits Line

As pointed out above, Proposition 8.3.1 requires the assumption that we restrict ourselves to the proﬁnite topology (i.e., that C is the pseudovariety of all ﬁnite groups), rather than a general pro-C topology. In this section we show that if the graph of groups (G,) considered in Proposition 8.3.1 corresponds to a free product of groups, then that proposition as well as Proposition 8.3.5 can be strengthened 8.4 Graph of a Free Product of Groups and the Tits Line 251 to the pro-C topology. To be more precise, in this section we are interested in a graph of abstract groups (G,)of the form

R2 1 1 R 1 1 3 . 1 . . Rn where we assume that each Ri is a residually C abstract group. Its fundamental abs = abs G group Π Π1 ( ,)is the free product R = R1 ∗···∗Rn, which is automatically residually C (cf. Gruenberg 1957, Theorem 4.1). Since the pro-C topology of R induces on each Ri its full pro-C topology (RZ, Corol- lary 3.1.6), one checks that RCˆ = (R1)Cˆ ···(Rn)Cˆ (the free pro-C product), so ¯ that RCˆ is the pro-C fundamental group of the graph (G,)of pro-C groups

(R1)Cˆ (R2)Cˆ 1 1 (R3) ˆ 1 1 C . 1 . .

(Rn)Cˆ Consider the corresponding standard graphs Sabs = Sabs(G,) and S = SC(G¯,). Explicitly (see Example 6.3.1) the sets of vertices and edges of the tree Sabs are n n abs . abs . V S = R/Ri and E S = R(ncopies of R), i=0 i=1 abs abs where R0 ={1}, and whose incidence maps d0,d1 : E(S ) → V(S ) are given by

abs d0(r) = rR0 = r r is in any copy of R in E S and

abs d1(r) = rRi, when r is in the ith copy of R in E S (i = 1,...,n). Similarly, the spaces of vertices and edges of the C-tree S are n n . . V(S)= RCˆ/(Ri)Cˆ and E(S) = RCˆ (n copies of RCˆ), i=0 i=1 252 8 Minimal Subtrees where (R0)Cˆ ={1}, and whose incidence maps d0,d1 : E(S) → V(S)are given by

d0(r) = r(R0)Cˆ = r r is in any copy of RCˆ in E(S) and

d1(r) = r(Ri)Cˆ, when r is in the ith copy of RCˆ in E(S) (i = 1,...,n). Then (see Sect. 6.5) Sabs is naturally embedded in S as a dense subgraph. In this case the explicit embedding Sabs −→ S is deﬁned as follows: for vertices

rRi → r(Ri)Cˆ (r ∈ R,i = 0,...,n), and for edges

abs r in the ith copy of R in E S → r in the ith copy of (R)Cˆ in E(S). abs Note that the R-stabilizers of edges in S are trivial, and similarly the RCˆ- stabilizers of edges in S are trivial. We start with an analogue of Proposition 8.2.4 in the case of the graph Sabs = abs S (R) associated with a free product R = R1 ∗···∗Rn.

Proposition 8.4.1 Let R1,...,Rn (n ≥ 2) be a ﬁnite collection of residually C abstract groups. Let H be a closed subgroup of the free product

R = R1 ∗···∗Rn (endowed with its pro-C topology). abs (a) Let S be the abstract standard tree associated to the free product R1 ∗···∗Rn and let T be an H -invariant subtree of Sabs such that H \T is finite. Then H\T = H¯ \T,¯ ¯ where H denotes the closure of H in RCˆ = (R1)Cˆ ···(Rn)Cˆ (free pro-C product), and T¯ is the closure of T in S, the standard C-tree associated to the pro-C product (R1)Cˆ ···(Rn)Cˆ. (b) Assume in addition that H is finitely generated and it is not contained in a abs conjugate of any Ri (i.e., H does not fix any vertex of S ). Then (b1) Sabs admits a unique minimal H -invariant subtree Dabs and its closure D in S is the unique minimal H¯ -invariant C-subtree of S; (b2) Sabs ∩ D = Dabs; and (b3) H \Dabs = H¯ \D is finite.

Proof (a) Consider the natural continuous map

T T¯ H¯ \T¯ 8.4 Graph of a Free Product of Groups and the Tits Line 253

Since its image is dense and H\T is ﬁnite, it induces an onto map H\T −→ H¯ \T.¯ Now, by Lemma 8.1.1, there exists an open subgroup V of R containing H such that τ : H\T −→ H\Sabs −→ V \Sabs is injective. Since V is open, one clearly has V \Sabs = V¯ \S: indeed, in this case the space of edges of these quotient graphs is a union of sets of right cosets of the \ = ¯ \ ¯ form V R V RCˆ, and the set of vertices is a union of V \R = V \RCˆ and sets of ¯ double cosets V \R/Ri = V \RCˆ/(Ri)Cˆ (i = 1,...,n). From the commutativity of the diagram = H\T H\Sabs V \Sabs V¯ \S

H¯ \T¯ one deduces that H\T −→ H¯ \T¯ is injective. abs (b) Observe that H is infinite. Say H = h1,...,hs. Fix a vertex v of S and consider the subgraph of Sabs s T = H[v,hiv], i=1 abs where[v,hiv] is the chain in S determined by v and hiv. Since the finite subgraph = s [ ] ∩ = ∅ = L i=1 v,hiv is obviously connected and since L hj L for every j 1,...,s, we deduce that HL= T is connected (see Lemma 2.2.4). Therefore T is a subtree of the tree Sabs. We deduce that the profinite subgraph s ¯ ¯ T = H[v,hiv] i=1 of S is connected, and hence a C-tree (see Lemma 2.1.7 and Proposition 2.4.3(b)). Note that H \T is finite, since it is contained in L. It follows from part (a) that H \T = H¯ \T¯ . Therefore, according to Lemma 8.2.1, T contains a unique minimal H -invariant subtree Dabs satisfying (b1)Ð(b3). By Lemma 8.1.2, D is also the unique minimal H -invariant subtree of Sabs.

When the subgroup H in the above proposition is cyclic, one gets more explicit results generalizing Proposition 8.3.1 so that now C is any extension-closed pseudovariety of finite groups. We omit the proofs of the several parts of the next proposition because some of the statements are special cases of those in Proposition 8.4.1 and the proofs of the remaining ones can be obtained mimicking those of the corresponding statements in Proposition 8.3.1, replacing ‘profinite topology’ with ‘pro-C topology’ and ‘profinite tree’ with ‘C-tree’. 254 8 Minimal Subtrees

Proposition 8.4.2 Let R1,...,Rn (n ≥ 2) be residually C abstract groups and let R = R1 ∗···∗Rn be their free product endowed with its pro-C topology. Let b ∈ R be a hyperbolic element (i.e., b is not in a conjugate of any Ri in R) such that b abs abs is closed. Let Lb be the corresponding Tits line in the standard tree S = S (R) C associated with that free product. Let S = S (RCˆ) be the standard C-tree associated with the free pro-C product

RCˆ = (R1)Cˆ ···(Rn)Cˆ. Then the following assertions hold. n n (a) b \Lb = b \Lb, for all natural numbers n. n abs (b) Lb is the unique minimal b -invariant C-subtree of S and Lb ∩ S = Lb. (c) If β ∈ b and βw ∈ Lb for some w ∈ Lb, then β ∈ b. abs (d) If β ∈ b− b, then βLb ∩ S (R) =∅. (e) Let {βλ | λ ∈ Λ} be a complete set of representatives of the cosets of b in b (a transversal). Then . Lb = βλLb. λ∈Λ

In other words, the abstract graphs βλLb are the distinct connected components of Lb considered as an abstract graph; in particular, Lb is its own connected component in Lb as an abstract graph. (f) Let N ={x ∈ R | xLb = Lb}. Then N is closed in the pro-C topology of R.

Similarly we obtain an improved analogue of Proposition 8.3.5 to the pro-C topology if we restrict ourselves to free products:

Proposition 8.4.3 Let R1,...,Rn (n ≥ 2) be residually C abstract groups and let R = R1 ∗···∗Rn be their free product. Let a,b ∈ R, and assume that a is hyperbolic (i.e., a is not contained in a conjugate in R of any of the factors Ri ). Assume that b is conjugate to a in RCˆ = (R1)Cˆ ···(Rn)Cˆ: a = bγ = γ −1bγ, where γ ∈ RCˆ. Then

(a) b is also hyperbolic and Lb = γ La, where La and Lb are the corresponding abs Tits lines in the standard tree S associated to the free product R1 ∗···∗Rn, while La and Lb denote the closures of La and Lb in the standard C-tree S associated to the free pro-C product (R1)Cˆ ···(Rn)Cˆ; (b) there exist x ∈ R such that x−1bx = a.

The proof of this proposition follows the same pattern as the proof of Proposi- tion 8.3.1; the necessary changes are these: in the proof of part (a) one appeals to Proposition 8.4.2(b) rather than to Proposition 8.3.1(b); and in the proof of part (b) one simply observes that in this case the stabilizer of any edge of S is trivial. 8.4 Graph of a Free Product of Groups and the Tits Line 255

Under the assumptions of this section, where we only consider free products of groups rather that more general fundamental groups, one can in fact generalize the above result and use ﬁnitely generated subgroups rather than elements. We make this precise in the next proposition.

Proposition 8.4.4 Let R1,...,Rn (n ≥ 2) be residually C abstract groups and let R = R1 ∗···∗Rn be their free product endowed with its pro-C topology. Let H1 and H2 be closed, ﬁnitely generated subgroups of R. Assume that H1 is not contained in a conjugate in R of any of the factors Ri . Suppose that H1 and H2 are conjugate in RCˆ = (R1)Cˆ ···(Rn)Cˆ: γ −1 H1 = H2 = γ H2γ, where γ ∈ RCˆ. Then H1 and H2 are conjugate in R: there exists some r ∈ R such that = r = −1 H1 H2 r H2r.

Proof First we claim that if M is a subgroup of R, then M is contained in a conju- ¯ gate in R of some Ri if and only if M is contained in an conjugate in RCˆ of some Rj (or equivalently, M fixes a vertex of Sabs if and only if M¯ fixes a vertex of S). In- −1 ¯ −1 deed, if M ≤ r Rir,forsomei = 1,...,n and some r ∈ R, then M ≤ r Rir. ¯ −1 Conversely, assume that M ≤ δ Rj δ,forsomej = 1,...,n and some δ ∈ RCˆ. Observe that if y1 and y2 are nontrivial elements of M that belong to different conjugates in R of the free factors R1,...,Rn, then y1y2 cannot be in a conjugate in R abs of any Ri . It follows from this observation that if M does not fix a vertex of S ,itis because there exists some y ∈ M acting freely on Sabs, i.e., y would be hyperbolic. But then y would act freely on S (see Lemma 8.3.2), contradicting our assumption. This proves the claim. It follows from this claim that H1 does not fix any vertex of S; therefore H2 = −1 γ H1γ cannot fix any vertex of S. Hence, using the claim again, H2 does not fix any vertex of Sabs. abs Thus Proposition 8.4.1 applies to both H1 and H2.LetDi be the unique min- abs = = abs imal Hi -invariant subtree of S (i 1, 2). Since D1 D1 is the unique min- −1 imal H1-invariant C-subtree of S and H2 = γ H1γ , one has that γD1 is the C = abs unique minimal H2-invariant -subtree of S.SoD2 γD1. By assumption D1 abs does not consist of only one vertex. Let e be an edge of D1 . Since, accord- \ abs = \ ˜ ∈ ing to Proposition 8.4.1, H2 D2 H2 D2, there exists some h2 H2 such that ˜ ∈ abs ⊆ abs abs ˜ h2γe D2 S . Since, by definition of S and S, h2γe and e have the same image in abs = R\S = RCˆ\S, ˜ there exists an r ∈ R such that re = h2γe. Since the RCˆ-stabilizer of e is trivial, one ˜ has h2γ = r ∈ R. Therefore, − γ h˜ 1r r = = 2 = = r H1 H2 H2 H2 H2 . 256 8 Minimal Subtrees

r Since H1 and H2 are closed (and hence so is H2 ), we deduce that (see Lemma 11.1.1(c), proved later in Chap. 11) = ∩ = ∩ r = r H1 R H1 R H2 H2 . Chapter 9 Homology and Graphs of Pro-C Groups

9.1 Direct Sums of Modules and Homology

Let Λ be a profinite ring and let M be a profinite Λ-module (see Sect. 1.7). The concept of a continuously indexed family {Mt | t ∈ T } of closed Λ-submodules of M indexed by a profinite space T is defined in a manner analogous to the definition of a continuously indexed family of closed subgroups of a profinite group (see Sect. 5.2). Similarly one has the concept of a sheaf of profinite Λ-modules (M,π,T) (see Sect. 5.1); in this case one requires not only that M is a sheaf of profinite abelian groups M(t), but also that the action map Λ × M −→ M that sends (λ, m) (λ ∈ Λ,m ∈ M(t)) to λm ∈ M(t) be continuous. Associated with these sheaves one has analogous definitions for internal and external free profinite products of Λ-modules, which in this case we term internal and external direct sums of Λ-modules. The notation we use for such sums is " " Mt or M. t∈T T

The results proved in Chap. 5 have natural analogues for direct sums of profinite modules. In this section we show that homology of profinite groups commutes with direct sum of modules indexed by a profinite space. In fact, we shall prove a more general result, namely that this is the case for the functors Torn(−, −) (see Sect. 1.9).

Theorem 9.1.1 Let Λ be a profinite ring. Let M #be a profinite right Λ-module = and N a profinite left Λ-module. Assume that M t∈T Mt is the direct sum of a continuously indexed family {Mt | t ∈ T } of profinite Λ-submodules Mt of M, where T is a profinite space. Then ∈ = Λ (a) for each t T and each n 0, 1, 2,...,Torn (Mt ,N) is canonically embed- Λ { Λ | ∈ } ded in Torn (M, N), and the collection Torn (Mt ,N) t T is a continuously Λ indexed family of profinite subgroups of Torn (M, N);

© Springer International Publishing AG 2017 257 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_9 258 9 Homology and Graphs of Pro-C Groups

(b) for each n = 0, 1, 2,..., " Λ = Λ ; Torn (M, N) Torn (Mt ,N) t∈T

Λ − (c) analogous results hold for the functors Torn (M, ).

Proof If T is finite, then the results follow from the additivity of the functor Λ − Torn ( ,N). Assume now that T is a general profinite space. Decompose M and T as in Theorem 5.3.4: there is a poset I and inverse systems {Ai,ϕij ,I} and {Ti,ρij ,I} of profinite Λ-modules Ai and finite discrete spaces Ti , respectively, such that " = = = M lim←− Ai,Tlim←− Ti, and Ai Ai,τ , ∈ ∈ i I i I τ∈Ti where each Ai,τ is a finite Λ-module, such that ϕij sends direct summands to direct summands and M = lim ∈ A , where t = (τ ) ∈ lim T = T . Since TorΛ(−,N)is t ←− i I i,τi i ←− i n additive, one has that " Λ = Λ Torn (Ai,N) Torn (Ai,τ ,N), τ∈Ti

∈ Λ − for each i I . Now, since Torn ( ,N)commutes with inverse limits (cf. RZ, Corol- lary 6.1.10), we have

Λ = Λ Λ = Λ Torn (M, N) lim←− Torn (Ai,N) and Torn (Mt ,N) lim←− Torn (Ai,τi ,N), i∈I i∈I

∈ = Λ for all t T , where (τi) t. Since Torn (Ai,τi ,N) is naturally embedded into Λ Torn (Ai,N), it follows that the canonical map Λ −→ Λ Torn (Mt ,N) Torn (M, N) is an embedding. Therefore the remaining statements in parts (a) and (b) now follow from Theorem 5.3.4.

Λ = ⊗ Specializing to the case Tor0 (M, N) M Λ N,wehave

Corollary 9.1.2 Complete tensoring commutes with direct sums over profinite spaces. Explicitly, let Λ be a profinite ring, M a profinite right Λ-module, N a profinite left Λ-module and T a profinite space. # = (a) If M t∈T Mt , then " M ⊗Λ N = (Mt ⊗Λ N). t∈T 9.2 Corestriction and Continuously Indexed Families of Subgroups 259 # = (b) If N t∈T Nt , then " M ⊗Λ N = (M ⊗Λ Nt ). t∈T

Let R be a commutative profinite ring. Let G be a profinite group and let [[ RG]] be the corresponding complete group ring (see Sect. 1.8). Let B be a profinite right [[ RG]] -module. Recall (see Sect. 1.10) that the n-th homology group of G with coefficients in B is defined as = [[ RG]] Hn(G, B) Torn (B, R), where one thinks of R as a left [[ RG]] -module with trivial G-action. We remark that Hn(G, B) is a profinite R-module. Then, as a consequence of Theorem 9.1.1,we obtain

Theorem 9.1.3 Let G be# a profinite group and let B be a profinite right [[ RG]] - = module. Assume that B t∈T Bt is the direct sum of a continuously indexed family {Bt | t ∈ T } of profinite [[ RG]] -submodules Bt of B, where T is a profinite space. Then

(a) for each t ∈ T and each n = 0, 1, 2,..., Hn(G, Bt ) is canonically embedded in Hn(G, B), and the collection {Hn(G, Bt ) | t ∈ T } is a continuously indexed family of R-submodules of Hn(G, B); (b) for each n = 0, 1, 2,..., " Hn(G, B) = Hn(G, Bt ). t∈T

9.2 Corestriction and Continuously Indexed Families of Subgroups

Let X be a profinite space. Let G be a profinite group and let R be a profinite ring. As indicated in Sects. 1.7 and 1.8, [[ RX]] denotes the free profinite R-module on the space X, and [[ RG]] denotes the complete group ring; while PMod([[ RG]] ) denotes the category of profinite [[ RG]] -modules. The following lemma is a slightly sharper version of Theorem 6.10.8(a) in RZ.

Lemma 9.2.1 Let K be a closed subgroup of a profinite group G and let R be a commutative profinite ring. Then the functor − ⊗[[ RK]] [[ RG]] sends projective profinite right [[ RK]] -modules to projective profinite right [[ RG]] - modules. 260 9 Homology and Graphs of Pro-C Groups

Proof Let B be a proﬁnite right [[ RK]] -module. According to Proposition 5.8.1 in RZ,

∼ B ⊗[[ RK]] [[ RG]] = B ⊗R R(K\G) , where the action of [[ RG]] on B ⊗[[ RK]] [[ RG]] is induced by the natural action on the right of [[ RG]] , and the action of [[ RG]] on B ⊗R [[ R(K\G)]] is the diagonal action. On the other hand, if we assume that B is a projective proﬁnite right [[ RG]] - module, then so is B ⊗R [[ R(K\G)]] by Proposition 5.8.3 in RZ.

Lemma 9.2.2 Let K be a closed subgroup of a profinite group G and let R be a commutative profinite ring. Let B be a profinite right [[ RK]] -module and let C be a profinite left [[ RG]] -module. Then for every n ≥ 0 the following assertions hold. (a) There exist a natural isomorphism

∼ : [[ RG]] ⊗ [[ ]] −→= [[ RK]] ϕn Torn B [[ RK]] RG ,C Torn (B, C).

(b) There exists a natural isomorphism

∼ = ϕn : Hn G, B ⊗R R(K\G) −→ Hn(K, B).

Hn(G, B ⊗R [[ R(K\G)]] )

εn

[[ RG]] ϕn Hn(G, B) = Torn (B, R)

Cor

Hn(K, B)

ε where εn is the map induced by the augmentation map [[ R(G/K)]] −→ R that ˜ ∈ ∈ = K sends every g G/K to 1 R, and Cor CorG is the usual corestriction homomorphism in homology (see Sect. 1.10). Assume now that B is a proﬁnite right [[ RG]] -module and let C be a proﬁnite left [[ RK]] -module. Then (d)

[[ RG]] [[ ]] ⊗ ∼ [[ RK]] ; Torn B, RG [[ RK]] C = Torn (B, C) (e) there exists a natural isomorphism

∼ : [[ RG]] −→= ϕn Torn B, R(G/K) Hn(K, B) 9.2 Corestriction and Continuously Indexed Families of Subgroups 261

such that the following diagram commutes

[[ RG]] Torn (B, [[ R(G/K)]] )

εn

[[ ]] ϕ RG n Hn(G, B) = Torn (B, R)

Cor

Hn(K, B)

Proof (a) The functor

− ⊗[[ RK]] [[ RG]] : PMod [[ RK]] −→ PMod [[ RG]] is exact, since [[ RG]] is a free [[ RK]] -module (see Sect. 1.8); moreover, it preserves projective modules according to Lemma 9.2.1. Therefore, for a fixed C, [[ RG]] the sequence of functors {Torn (− ⊗[[ RK]] [[ RG]] ,C)}n is universal on the category of profinite right [[ RK]] -modules (see Sect. 1.9). On the other hand, the se- [[ RK]] quence {Torn (−,C)}n is also universal on the category of profinite right [[ RK]] - modules. Therefore, to prove the assertion it suffices to check it in dimension 0. But this is clear: one has natural isomorphisms

[[ RG]] ⊗ [[ ]] = ⊗ [[ ]] ⊗ =∼ ⊗ Tor0 B [[ RK]] RG ,C B [[ RK]] RG [[ RG]] C B [[ RK]] C. (b) We observe ﬁrst that

∼ B ⊗R R(K\G) = B ⊗[[ RK]] [[ RG]]

(cf. RZ, Proposition 5.8.1). Hence (b) follows from (a) by putting C = R. (c) Again it sufﬁces to check the commutativity of the diagram for n = 0, in which case the result is easily veriﬁed. Parts (d) and (e) are proved similarly.

Let F ={Gt | t ∈ T } be a continuously indexed family of closed subgroups of a profinite group G, where T is a profinite space. Let G/F = T × G/ ∼= G/Gt t∈T be the quotient space defined in Proposition 5.2.3.Letp : G/F −→ T be the map given by p(gGt ) = t(g∈ G, t ∈ T). One easily checks that p is continuous; and ob- −1 viously p (t) = G/Gt . On the other hand G acts continuously on the left of T ×G by multiplication on the second component, and this action is compatible with the equivalence relation ∼ defined in Proposition 5.2.3. This induces a continuous left 262 9 Homology and Graphs of Pro-C Groups action of G on G/F given by x(gGt ) = xgGt (g, x ∈ G). Hence [[ R(G/F)]] and [[ R(G/Gt )]] have in a natural way the structure of left [[ RG]] -modules. Assume that R is a profinite quotient ring of ZCˆ = Zπˆ , where π = π(C). It then follows from (the equivalent of) Example 5.1.3(b) and Proposition 5.5.4 that " R(G/F) = R(G/Gt ) , t∈T as [[ RG]] -modules. Therefore, by Theorem 9.1.1, " [[ RG]] F = [[ RG]] Torn B, R(G/ ) Torn B, R(G/Gt ) , t∈T for every profinite right [[ RG]] -module B. By Lemma 9.2.2(e), one can identify [[ RG]] Hn(Gt ,B)(t ∈ T)with a closed R-submodule of Torn (B, [[ R(G × T/F)]] ) via ∼ [[ RG]] = ϕn : Torn (B, [[ R(G/Gt )]] ) −→ Hn(Gt ,B)and the embedding

[[ RG]] −→ [[ RG]] F Torn B, R(G/Gt ) Torn B, R(G/ ) . Hence, " [[ RG]] F = Torn B, R(G/ ) Hn(Gt ,B). t∈T Now, the augmentation [[ RG]] -homomorphism ε :[[R(G/F)]] −→ R that sends each element of G/F to 1 ∈ R, induces a continuous R-homomorphism " = [[ RG]] F −→ [[ RG]] = Hn(Gt ,B) Torn B, R(G/ ) Torn (B, R) Hn(G, B), t∈T

F whichwecallcorestriction and denote by CorG . By the commutativity of the diagram in Lemma 9.2.2(e), this map restricts to the usual corestriction of groups

Gt : −→ ∈ CorG Hn(Gt ,B) Hn(G, B) (t T). We collect these facts in the following proposition.

Proposition 9.2.3 Assume that R is a profinite quotient ring of ZCˆ. Let F ={Gt | r ∈ T } be a continuously indexed family of closed subgroups of a profinite group G, where T is a profinite space. Let B be a profinite right [[ RG]] -module. Then

(a) for each t ∈ T and n ≥ 0, Hn(Gt ,B) can be naturally identiﬁed with the R- [[ RG]] [[ ]] [[ RG]] [[ F ]] submodule Torn (B, R(G/Gt ) ) of#Torn (B, R(G/ ) ); (b) there is a naturally deﬁned direct sum t∈T Hn(Gt ,B) over the space T , and a continuous R-homomorphism " F : −→ CorG Hn(Gt ,B) Hn(G, B), t∈T 9.2 Corestriction and Continuously Indexed Families of Subgroups 263

called corestriction, which is induced by the usual corestriction homomor- Gt : −→ ∈ phisms of groups CorG Hn(Gt ,B) Hn(G, B) (t T).

Theorem 9.2.4 Assume that R is a profinite quotient ring of ZCˆ. Let F ={Gt | t ∈ T } be a continuously indexed family of closed subgroups of a profinite group G, where T is a profinite space. Suppose that for some n ≥ 0 the corestriction map " F : −→ CorG Hn(Gt ,B) Hn(G, B) t∈T is an isomorphism for every profinite right [[ RG]] -module B. −1 (a) Let r, s ∈ T . If, for some g ∈ G, Gs ∩ gGr g contains a nontrivial finite group, then r = s and g ∈ Gr . −1 (b) Let H be a finite subgroup of G. Then H ≤ gGr g , for some g ∈ G and some r ∈ T .

Proof Let K be a closed subgroup of G and let p ∈ π(C). Denote by Fp the ﬁeld with p elements. Then we have isomorphisms of proﬁnite right [[ FpG]] -modules (cf. RZ, Proposition 5.8.1)

\ =∼ ⊗ \ =∼ ⊗ [[ ]] Fp(K G) Fp Fp Fp(K G) Fp [[ FpK]] FpG .

Therefore, using Shapiro’s Lemma (see Sect. 1.10), we have

∼ Hn G, Fp(K\G) = Hn(K, Fp).

So, by our hypotheses, " ∼ Hn Gt , Fp(K\G) = Hn(K, Fp). (9.1) t∈T

Let t ∈ T . The group Gt acts continuously on the right of the (compact) space K\G. The quotient space under this action is the (compact) space K\G/Gt of double cosets. Denote by

δ : K\G −→ K\G/Gt the corresponding quotient map. Then, using (the analogue of) Example 5.1.3(b), Proposition 5.5.4 and Theorem 9.1.3, we have that " ∼ −1 Hn Gt , Fp(K\G) = Hn Gt , Fp δ (σ ) σ ∈K\G/Gt " −1 = Hn Gt , Fp δ (σ ) . (9.2) σ ∈K\G/Gt 264 9 Homology and Graphs of Pro-C Groups

−1 Let σ ∈ K\G/Gt ; then σ = Kgσ Gt ,forsomegσ ∈ G. Note that δ (σ ) is the \ ∩ −1 Gt -orbit in K G of Kgσ . Since the Gt -stabilizer of Kgσ is Gt gσ Kgσ ,wehave a continuous bijection of Gt -spaces (cf. Bourbaki 1989, Sect. III.5) ∩ −1 \ −→ −1 Gt gσ Kgσ Gt δ (σ ). This bijection is in fact a homeomorphism since the spaces involved are compact. Therefore, using this and Shapiro’s Lemma (as above),

−1 ∼ ∩ −1 \ Hn Gt , Fp δ (σ ) = Hn Gt , Fp Gt gσ Kgσ Gt

∼ ∩ −1 = Hn Gt gσ Kgσ , Fp . (9.3) We are now in a position to prove part (a) of the theorem. Choose a prime −1 number p and a cyclic subgroup K of Gs ∩ gGr g of order p.SoK ≤ Gs and −1 g Kg ≤ Gr ; therefore, using a standard computation for the homology of a cyclic group (see, for example, Serre 1968, Chap. VIII, ¤4), one has

−1 −1 ∼ Hn Gr ∩ g Kg,Fp = Hn g Kg,Fp = Fp. (9.4)

It follows from (9.2) and (9.3) that Hn(Gr , [[ Fp(K\G)]] ) = 0. Similarly #Hn(Gs, [[ Fp(K\G)]] ) = 0. On the other hand, we deduce from (9.1) that [[ \ ]] =∼ = [[ \ ]] =∼ t∈T Hn(Gt , Fp(K G) ) Fp. Thus r s and Hn(Gr , Fp(K G) ) Fp. Finally, to show that g ∈ Gr , consider the double cosets σ = KgGr and τ = −1 K1Gr .From(9.4) and (9.3)wehaveHn(Gr , [[ Fp(δ (σ ))]] ) = 0. Similarly −1 Hn(Gr , [[ Fp(δ (τ))]] ) = 0. We then deduce from (9.2) that σ = τ , i.e., KgGr = K1Gr .Sog ∈ Gr , since K ≤ Gr . This concludes the proof of part (a). We prove part (b) by induction on the order of H . Assume ﬁrst that H is cyclic of prime order p. Then H (H, F ) =∼ F . Hence we deduce from (9.1) that ∼ n p p Hn(Gr , [[ Fp(H \G)]] ) = Fp,forsomer ∈ T . It follows from (9.2) and (9.3) that

−1 Hn Gr ∩ g Hg,Fp = 0,

−1 for some g ∈ G. Therefore, Gr ∩ g Hg = 1. Since H has prime order, one has that −1 H ≤ gGr g in this case. Next assume that H is a nontrivial ﬁnite group whose order is not a prime. We claim that H contains a subgroup L and an element h such that H = h, L and L ∩ hLh−1 = 1. To see this, let M be a maximal subgroup of H .IfM ∩ hMh−1 = 1 for some h ∈ H − M, then put L = M.IfM ∩ hMh−1 = 1 for all h ∈ H − M, then H is a Frobenius group and it has the form H = N M, where N is its Frobenius kernel (cf. Huppert 1967, Sect. V.8). Choose L to be a maximal subgroup of H containing N and let h ∈ H − L. Then L ∩ hLh−1 ≥ N and obviously H = h, L, proving the claim. By the induction hypothesis, there exists some r ∈ T and some −1 g ∈ G such that L ≤ gGr g . Then

−1 −1 −1 −1 1 = L ∩ hLh ≤ gGr g ∩ hgGr g h .

−1 −1 −1 It follows from part (a) that g hg ∈ Gr , and so h ∈ gGr g . Thus H ≤ gGr g . 9.3 The Homology Sequence of the Action on a Tree 265

9.3 The Homology Sequence of the Action on a Tree

In this section R is a quotient ring of Zπˆ , where π = π(C).

Let T be a π-tree and let G beapro-C group that acts continuously on T .By deﬁnition (see Proposition 2.4.2) the sequence

∗ d ε 0 −→ R E (T ), ∗ −→ R V(T) −→ R −→ 0 (9.5) is exact, where E∗(T ) = T/V(T). The action of G on T induces continuous actions of G on the space V = V(T)and on the pointed space (E∗, ∗) = (E∗(T ), ∗). These actions in turn make [[ RV ]] and [[ R(E∗, ∗)]] into proﬁnite left [[ RG]] -modules (cf. RZ, Sect. 5.7). Let B be a proﬁnite right [[ RG]] -module. Then, associated with the [[ RG]] sequence of functors {Torn (B, −)}n≥0 and the short exact sequence (9.5), we obtain a long exact sequence (see Sect. 1.9)

···→ [[ RG]] →δ [[ RG]] ∗ ∗ Tor2 (B, R) Tor1 B, R E (T ),

[[ RG]] [[ RG]] δ ∗ → Tor B,[[ RV ]] → Tor (B, R) → B ⊗[[ RG]] R E (T ), ∗ 1 1 → B ⊗[[ RG]] R V(T) → B ⊗[[ RG]] R −→ 0. (9.6) We refer to (9.6)asthehomology sequence of the action of G on T . Denote by

∗ ∗ ηV : V −→ G\V and ηE∗ : E , ∗ −→ G\E , ∗ the quotient maps. Let

∗ ∗ σV : G\V −→ V and σE∗ : G\E , ∗ −→ E , ∗ be sections (not necessarily continuous) of ηV and ηE∗ , respectively.

Lemma 9.3.1 We continue with the notation above. If m ∈ V(T)∪ E(T ), let m denote its canonical image in (G\V(T))∪ (G\E∗(T )). For each n = 0, 1,..., the following assertions hold. (a) [[ RG]] = Torn (B, R) Hn(G, B). (b) " [[ RG]] [[ ]] = [[ RG]] −1 Torn B, RV Torn B, R ηV (v) v∈G\V and " [[ RG]] ∗ ∗ = [[ RG]] [[ ]] Torn B, R E , Torn B, RAe , e∈G\E∗ 266 9 Homology and Graphs of Pro-C Groups

where −1 η ∗ (e), if e = ∗, A = E e −1 ∗ ∗ =∗ (ηE∗ ( ), ), if e .

[[ RG]] −1 =∼ Torn B, R ηV (v) Hn(Gv,B), ∗ [[ RG]] ∼ (d) For e ∈ G\E , one has Torn (B, [[ RAe]] ) = 0, if e =∗, while [[ RG]] ∼ Torn (B, [[ RAe]] ) = Hn(Ge,B), if e = ∗.

Proof Part (a) is just the deﬁnition of Hn(G, B). Interpreting the free abelian pro-C group [[ RV ]] as a free product in the category of abelian pro-C groups (see Exam- ple 5.1.3) we have (see Proposition 5.5.4) " [[ ]] = −1 RV RηV (v) . v∈G\V So the ﬁrst part of (b) follows from Theorem 9.1.1; and the second part is proved in ∈ −1 = · an analogous manner. To prove part (c), let v V . We observe that ηV (v) G v. On the other hand, there exists a homeomorphism −→ · = −1 G/Gv G v ηV (v), given by gGv → gv (it is a continuous bijection and the spaces G/Gv and G · v are compact). Furthermore, this homeomorphism is compatible with the action of G:

g (gGv) → g (gv) g,g ∈ G, v ∈ V , [[ −1 ]] [[ ]] [[ ]] so that R(ηV (v)) and R(G/Gv) are isomorphic as RG -modules. There- fore, using Lemma 9.2.2(e),

[[ RG]] −1 =∼ [[ RG]] =∼ Torn B, R ηV (v) Torn B, R(G/Gv) Hn(Gv,B), as asserted. = ∗ = −1 Finally, we prove part (d). Note that if e , then Ae ηE∗ (e) is homeomorphic with the G-orbit G · e of e in E because (G · e) ∩ V =∅. We can then apply the argument used above to obtain

[[ RG]] [[ ]] ∼ Torn B, RAe = Hn(Ge,B), as desired. =∗ −1 ∗ ={∗} If, on the other hand, e , note that ηE∗ ( ) ; then by part (b),

[[ RG]] [[ ]] =∼ [[ RG]] −1 ∗ ∗ Torn B, RAe Torn B, R ηE∗ ( ),

= [[ RG]] {∗} ∗ = Torn B, R , 0, because [[ R({∗}, ∗)]] is just the free R-module on the pointed space ({∗}, ∗). 9.4 MayerÐVietoris Sequences 267

Let p be a ﬁxed prime number. Next we record the following consequences for the cohomological p-dimension (see Sect. 1.11) of a proﬁnite group acting on a C-tree.

Corollary 9.3.2 Let G be a pro-C group that acts on a C-tree T . Then the following assertions hold. (a) For every prime number p ∈ π(C), cdp(G) ≤ sup cdp(Gv), cdp(Ge) v ∈ V(T),e∈ E(T ) + 1.

(b) If G acts freely on T , then G is a projective proﬁnite group. (c) Let Cp ∈ C and assume that G is a pro-p group that acts freely on T . Then G is a free pro-p group.

Proof (a) follows from the exactness of the sequence (9.6) and Lemma 9.3.1.IfG acts freely on T , then Gm = 1, for every m ∈ T ; so part (b) follows from (a) (see Sect. 1.11). An alternative proof can be deduced directly from the exactness of the sequence (9.5): if p ∈ π(C), one deduces that the sequence

∗ d ε 0 −→ Fp E (T ), ∗ −→ Fp V(T) −→ Fp −→ 0 is exact. But this is an [[ FpG]] -free resolution of the [[ FpG]] -module Fp (cf. RZ, Proposition 5.7.1; note that this proposition is stated only for nonpointed spaces, but a similar result and an analogous proof is valid for pointed spaces); hence cdp(G) ≤ 1 and therefore G is projective (see Sect. 1.11). Part (c) follows from the fact that projective pro-p groups are free pro-p.

See Theorem 4.1.2 for an alternative proof of part (b) of the above corollary.

9.4 MayerÐVietoris Sequences

In this section C is an extension-closed pseudovariety of ﬁnite groups.

Let G be a pro-C group that acts continuously on a C-tree T . As pointed out earlier this action induces continuous actions of G on the space V(T) and on the pointed space (E∗(T ), ∗), where E∗(T ) = T/V(T). Assume that the natural quotient map η : T −→ G\T admits a continuous section

σ : G\T −→ T.

Then we can write the long exact sequence (9.6) in terms of the homology groups of stabilizer groups of the action, as we describe presently. First observe that σ induces continuous sections

∗ ∗ σV : G\V(T)−→ V(T) and σE∗ : G\E (T ), ∗ −→ E (T ), ∗ 268 9 Homology and Graphs of Pro-C Groups of the quotient maps

∗ ∗ ηV : V(T)−→ G\V(T) and ηE∗ : E (T ), ∗ −→ G\E (T ), ∗ , respectively. The continuity of σV and σE∗ together with Lemma 5.2.2 imply that the families of closed subgroups of G ∗ F = ∈ \ F ∗ = ∈ \ V GσV (v) v G V(T) and E GσE∗ (e) e G E (T ) are continuously indexed by G\V(T)and G\E∗(T ), respectively. By Lemma 5.2.4, there are homeomorphisms

! ! ∗ ψV : G/FV −→ V(T) and ψE∗ : G/FE∗ −→ E (T ) such that

= · = −1 ∈ \ ψV (G/GσV (v)) G σV (v) ηV (v) v G V(T) and

−1 ∗ ∗ = · = ∗ ∈ \ ψE (G/GσE∗ (e)) G σV (e) ηE (e) e G E (T ) . Thus, using Lemma 9.3.1 and Lemma 9.2.2, the long exact sequence (9.6) can be written as " ··· → −→δ Hi+1(G, B) Hi(GσE∗ (e),B) e∈G\E∗(T ) " Cor −→ d(i) −→ Hi(GσV (v),B) Hi(G, B) v∈G\V(T) " −→ δ →··· Hi−1(GσE∗ (e),B) e∈G\E∗(T ) where Cor is the corestriction map deﬁned in Sect. 9.2. An instance of the above situation is the following. Let Γ be a proﬁnite graph such that V(Γ) is clopen, and let (G,Γ) be an injective graph of pro-C groups over Γ .Let(ν, ν ) be a universal specialization of (G,Γ) in its fundamental pro-C = C G = G C = group Π Π1 ( ,Γ), and put Π(m) ν( (m)). Then the -standard graph S C S (G,Γ)of this graph of groups is a C-tree (see Corollary 6.3.6). We then have that \ = \ = : −→ = . Π V(S) V(Γ), Π E(S) E(Γ ), and that σ Γ S m∈Γ Π/Π(m), given by m → 1Π(m) (m ∈ Γ), is a continuous section of the projection p : ∼ S −→ Γ (see Lemma 6.3.2). Moreover, Πσ(m) = Π(m)= G(m), for every m ∈ Γ . Therefore we have proved the following

Theorem 9.4.1 (MayerÐVietoris sequence of a graph of groups) We continue with the assumptions and notation of the paragraph above. Let the proﬁnite ring R be a 9.4 MayerÐVietoris Sequences 269 quotient ring of ZCˆ. For every proﬁnite right [[ RΠ]] -module B, there exist a long exact sequence associated with the graph of pro-C groups (G,Γ): " δ ··· → Hi+1(Π, B) −→ Hi G(e), B e∈E(Γ ) " " d Cor δ −→ (i) Hi G(v), B −→ Hi(Π, B) −→ Hi−1 G(e), B →··· v∈V(Γ) e∈E(Γ ) that we term the Mayer–Vietoris sequence of the graph of groups (G,Γ).

Corollary 9.4.2 (MayerÐVietoris sequence of an amalgamated product) Let G = G1 H G2 be a proper amalgamated free pro-C product of the pro-C groups G1 and G2 amalgamating the common closed subgroup H , and let R be a quotient ring of ZCˆ. Then for every proﬁnite right [[ RG]] -module B, there exist a long exact sequence associated with the amalgamated product G = G1 H G2:

δ ···→Hi+1(G, B) −→ Hi(H, B)

d(i) Cor δ −→ Hi(G1,B)⊕ Hi(G2,B)−→ Hi(G, B) −→ Hi−1(H, B) →···

C Next we apply! the above results to the case of a pro- group G that is the free pro- C = C { | ∈ } product G x∈X Gx of a family of closed subgroups Gx x X continuously indexed by a proﬁnite space X.LetR be a proﬁnite ring and let G beapro-C group. If (( I G )) is the augmentation ideal of the complete group ring [[ RG]] , then one has that H0(G, B) = BG, where BG = B/B((IG)) (see Sect. 1.10).

{ | ∈ } C Theorem 9.4.3 Let Gx x X be a family of closed subgroups of a! pro- group = C G continuously indexed by a proﬁnite space X and assume that G x∈X Gx is their free pro-C product. Let B be a right [[ RG]] -module, where R is a proﬁnite quotient ring of ZCˆ. Then (a) " Cor : Hi(Gx,B)−→ Hi(G, B) x∈X is an isomorphism for i ≥ 2 and a monomorphism for i = 1; and (b) there is an exact sequence " " Cor δ 0 → H1(Gx,B)−→ H1(G, B) → B((IGx)) → B → BG → 0, x∈X x∈X

where BG = B/B((IG)). ! = C C Proof We interpret G x∈X Gx as the fundamental pro- group of a graph of groups over a graph T = T(X) as in Example 6.2.3(b). The vertex groups are the 270 9 Homology and Graphs of Pro-C Groups groups Gx (x ∈ X = V(T)) together with one trivial group Gω = 1, and the edge groups Ge (e ∈ E(T ) ={(ω, x) | x ∈ X}) are all trivial. So Hi(Ge,B)= 0ifi ≥ 1, for all the edge groups. Therefore, part (a) follows from the exactness of the long sequence in Theorem 9.4.1. To prove part (b) we note ﬁrst that " " " H0(Ge,B)= H0(1,B)= B e∈E(T ) E(T ) E(T ) and " " = ⊕ H0(Gx,B) BGx B. X∪{ω} x∈X Hence from the last six terms of the sequence of Theorem 9.4.1 we obtain the exact sequence, " " " → −→Cor −→δ −→α ⊕ −→β → 0 H1(Gx,B) H1(G, B) B BGx B BG 0, x∈X E(T ) x∈X + ∈ = + ∈ = where β sends b B((IGx)) B/((IGx)) BGx to b B((IG)) B / (( I G )) BG, and β(B) = BG; while, if b belongs to the e copy of B = H0(Ge,B)= H0(1,B), then

= + − ∈ ⊕ α(b) b B((IGx)) , b BGx B.

Since β(B) = BG, we deduce that the sequence " Cor δ −1 α β 0 → H1(Gx,B)−→ H1(G, B) −→ α {0}+B −→ B −→ BG → 0 x∈X is exact. Next observe that Im(δ) is contained in the closure of the set of subgroups Ker(α|B ) for all copies of B (this is clear if X is ﬁnite, and in general it follows by an inverse limit argument). Finally, note that α−1({0}+B) intersects each copy of B in B((IGx)) ( x ∈ X). Let us denote the closure of all these intersections by L; then B ∩ L = (B ∩ L) = B((IGx)) E(T ) E(T ) X # = is closed. So L X B((IGx)) by Theorem 5.5.6. This proves part (b).

9.5 Homological Characterization of Free Pro-p Products

Let F ={Gx | x ∈ X} be a continuously indexed family of closed subgroups of a pro-p group G. This section contains a useful homological criterion to determine whether G is the free pro-p product of the family F. 9.5 Homological Characterization of Free Pro-p Products 271

Theorem 9.5.1 Let p be a ﬁxed prime number. For a proﬁnite space X, let F ={Gx | x ∈ X} be a continuously indexed family of closed subgroups of a pro-p group G. The following conditions are equivalent: (a) There exists a free pro-p subgroup F of G such that G is the free pro-p product G = Gx F. x∈X (b) The corestriction map " = F : −→ Cor CorG Hi(Gx, Fp) Hi(G, Fp) x∈X

is injective for i = 1 and surjective for i = 2, where Fp is the additive group of the ﬁeld with p elements with trivial G-action. ! Proof Throughout the proof indicates ‘free pro-p product’. In one direction the result follows from Theorem 9.4.3, once one observes that H2(F, Fp) = 0, since F F is free pro-p. To prove the converse, assume that CorG is injective in dimension 1 and surjective in dimension 2. ∼ It is known (see Sect. 1.10) that H1(K, Fp) = K/Φ(K) for every pro-p group K, where Φ(K) is the Frattini subgroup of K. Furthermore, this isomorphism is natural, in the sense that, under this identiﬁcation, if K1 ≤ K, then the image of H1(K1, Fp) in H1(K, Fp) under the corestriction map is ∼ K1Φ(K)/Φ(K) = K1/L1 ∩ Φ(K) = K1/Φ(K1).

Deﬁne a sheaf (G,π,X)of pro-p groups by G = (x, g) ∈ X × G g ∈ Gx , ! : G −→ = G where π X is the natural projection. Let L X , and let

ϕ : G −→ G X be the continuous homomorphism induced by the projection G −→ G. Denote by B the closed subgroup ϕ (L )Φ(G)/Φ(G) of G/Φ(G). Then G/Φ(G) = B × C, for some closed subgroup C of G/Φ(G) (cf. RZ, Proposition 2.8.16); furthermore, = C Y Fp, for some set Y (we think of Y as a set of generators of C converging to 1). Let σ : G/Φ(G) −→ G be a continuous section of the canonical epimorphism G −→ G/Φ(G) (see Sect. 1.3). Put Y = σ(Y ). Then Y is a subset of G converging to 1. Furthermore, G is topologically generated by ϕ (L ) and Y .LetF = F(Y)be a free pro-p group on 272 9 Homology and Graphs of Pro-C Groups the set Y converging to 1; and let ϕ : F −→ G be the continuous epimorphism that sends Y ⊆ F to Y ⊆ G identically. Deﬁne L = L F (free pro-p product), and let

ϕ : L −→ G be the continuous epimorphism induced by ϕ and ϕ . It follows that ϕ is an epimorphism. We need to prove that ϕ is an isomorphism. To do this it sufﬁces to show that the coinﬂation map

Coinf : Hi(L, Fp) −→ Hi(G, Fp) determined by ϕ is a monomorphism for i = 1 and an epimorphism for i = 2(cf. RZ, Proposition 7.2.7 for the dual version of this criterion). This in turn follows from our hypotheses. Indeed, we note ﬁrst that there are commutative diagrams # ⊕ x∈X Hi(Gx, Fp) Hi(F, Fp) G ψi L ψi

Hi(L, Fp) Hi(G, Fp), Coinf

L G F F F where ψi and ψi are the maps CorL and CorG , respectively, and where is the family {Gx | x ∈ X}∪{F } interpreted as a subfamily of subgroups of L and of G, L L respectively. Now, ψ1 and ψ2 are isomorphisms by Theorem 9.4.3 (in dimension 1 because, in addition, the module Fp has trivial L-action). Hence we need to show G G = G = F that ψ2 is a surjection and ψ1 is an injection. Since H2(F, Fp) 0, ψ2 CorG , and hence a surjection by hypothesis. Finally, = F ⊕ ψ1 CorG idC = ⊕ F is a monomorphism, because on the one hand, H1(G, Fp) B C,CorG has G its image in B and it is injective by hypothesis; and on the other hand, ψ1 maps H1(F, Fp) identically onto C.

9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem

Throughout this section C is a pseudovariety of ﬁnite groups closed under extensions with abelian kernel.

In this section we describe the structure of certain pro-p groups that act on a C- tree, and as a consequence we prove an analogue of the Kurosh subgroup theorem for certain pro-p subgroups of free pro-C products. Recall that a proﬁnite group is called second-countable if its topology admits a countable base of clopen sets, or equivalently, if G admits a countable set of generators converging to 1 (see Sect. 1.3). 9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem 273

Theorem 9.6.1 Let p ∈ π = π(C) and let G be a pro-p group that acts continuously on a C-tree T so that Ge = 1, for every edge e ∈ E = E(T ). Put V = V(T) and QV = G\V . Let

σ : QV = G\V −→ V beasection(not necessarily continuous) of the canonical projection map η : V −→ QV = G\V .

(a) If the collection of G-stabilizers {Gσ(v) | v ∈ QV } is continuously indexed by QV , then G is a free pro-p product G = Gσ(v) F, v∈QV where F is a free pro-p group. (b) Assume that G is second-countable. Then a section σ can be chosen so that the family of subgroups {Gσ(v) | v ∈ QV } of G is continuously indexed by QV . (c) Assume that V = V(T)is second-countable as a proﬁnite space. Furthermore, assume that W is a closed subset of V such that the restriction η|W of η to W is injective. Then σ can be chosen so that W ⊆ σ(QV ) and the family of subgroups {Gσ(v) | v ∈ QV } of G is continuously indexed by QV .

Proof (a) Assume that the collection of subgroups {Gσ(v) | v ∈ QV } of G is continuously indexed by QV , where σ is a certain section of η = ηV .LetR be a proﬁnite quotient ring of ZCˆ. Since Ge = 1, for all e ∈ E, it follows from Lemma 9.3.1(b), (d) that,

[[ RG]] ∗ ∗ = ≥ Torn B, R E , 0,(n1), for every proﬁnite right [[ RG]] -module B. Therefore by the exactness of sequence (9.6), we have that the natural continuous homomorphism

[[ RG]] [[ ]] −→ [[ RG]] Torn B, RV Torn (B, R), induced by the map that sends V to 1, is injective for n = 1 and an isomorphism for n = 2. Therefore, using Lemma 9.3.1, the continuous homomorphism " [[ RG]] −1 −→ = [[ RG]] Torn B, R ηV (v) Hn(G, B) Torn (B, R), (9.7) v∈G\V

−1 ∈ \ = obtained by sending ηV (v) to 1 (v G V), is injective for n 1 and an isomorphism for n = 2. [[ RG]] [[ −1 ]] Next identify Torn (Z/pZ, R(ηV (v)) ) with Hn(GσV (v),R) as in Lem- mas 9.3.1(c) and 9.2.2(e). Then, according with the deﬁnition of the map Cor, (9.7) becomes " : −→ Cor Hn(GσV (v),R) Hn(G, R). v∈QV 274 9 Homology and Graphs of Pro-C Groups

This is an injection if n = 1 and an isomorphism for n = 2. Since the family

{ | ∈ } GσV (v) v QV is continuously indexed, it follows from Theorem 9.5.1 that there exists a free pro-p subgroup F of G such that G is the free pro-p product = G GσV (v) F, v∈QV proving part (a).

(b) Put V ={v ∈ V | Gv = 1}, and deﬁne X = V to be the topological closure of V in V . Note that G acts on X.Let

η = ηV : V −→ G\V = QV and ηX : X −→ G\X = QX denote the quotient maps. For v ∈ V and x ∈ X,letv and x denote the corresponding images in G\V and G\X, respectively. = ∈ ∩ = Since Ge 1, for every e E(T ), we deduce that Gv1 Gv2 1, whenever v1 = v2 are in V = V(T)(see Corollary 4.1.6). Hence, by Proposition 5.4.2, there exists a second-countable proﬁnite G-space X˜ and an epimorphism

μ : X −→ X˜

of G-spaces which is an injection on V , such that Gx = Gμ(x), for every x ∈ X. ˜ : ˜ −→ \ ˜ Since X is second-countable, the quotient map ηX˜ X G X admits a contin- ˜ ˜ ˜ uous section σ ˜ : G\X −→ X (see Sect. 1.3). It follows that {G ˜ |˜x ∈ G\X} X σX˜ (x) is a family of subgroups of G continuously indexed by G\X˜ (this follows from Lemma 5.2.2 and the continuity of σX˜ ).

ηV V G\V

σV

ηX X = V G\X

σX μ μ

ηX˜ X˜ G\X˜

σX˜

Let μ : G\X −→ G\X˜ be the continuous map induced by μ. Since μ is continuous, the family {G | x ∈ G\X} of closed subgroups of G is continuously (σX˜ μ)(x) indexed by G\X. 9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem 275

Choose σX : G\X −→ X to be a section (not necessarily continuous) of ηX such that μσ = σ ˜ μ. By Proposition 5.4.2 (c), G = G , for every x ∈ G\X. X X σX(x) (σX˜ μ)(x) Therefore {G | x ∈ G\X}={G | x ∈ G\X} σX(x) (σX˜ μ)(x) is a family of closed subgroups of G continuously indexed by G\X. Extend σX arbitrarily to a section σ = σV : G\V −→ V . We claim that { | ∈ = \ } GσV (v) v QV G V is a family of closed subgroups of G continuously indexed by G\V . To see this we must show that if U is an open subset of G, then ={ | ⊆ } QV (U) v GσV (v) U = \ ={ | ⊆ } = is open in QV G V . We know that QX(U) x GσX(x) U is open in QX G\X. Hence there exists an open subset S of QV such that S ∩ QX = QX(U). ∈ − = Observe that if v QV QX, then GσV (v) 1. It follows that

QV (U) = S ∪ (QV − QX) is open in QV , proving the claim. This proves part (b). (c) Since V is second-countable, the section σ can be chosen to be continuous and such that W ⊆ σ(Q ) (cf. RZ, Lemma 5.6.7). By Lemma 5.2.2 {G ˜ |˜x ∈ V σX˜ (x) G\X˜ } is a family of subgroups of G continuously indexed by G\X˜ .

Next we have the following analogue of the classical Kurosh subgroup theorem in the context of free pro-p products (see also Theorem 7.3.1, where somewhat more precise information is obtained for open subgroups of free products of ﬁnitely many pro-C groups). See Theorem 10.7.4 for a counterexample of a possible general analogue of the Kurosh theorem for free pro-p products of pro-p groups.

Theorem 9.6.2 Let C be an extension-closed pseudovariety of ﬁnite groups and let p be a prime number such that Cp ∈ C. Let H beapro-C group together with a collection F ={Hz | z ∈ Z} of closed subgroups continuously indexed by a proﬁnite space Z, and assume that H is their free pro-C product:

H = Hz. z∈Z Let G be a second-countable pro-p subgroup of H . ∈ { } (a) Then, for each z Z, there is a complete set hi,z i∈Iz of representatives of the \ { ∩ −1} double cosets G H/Hz such that G hi,zHzhi,z i,z is a continuously indexed family of subgroups of G, and G is a free pro-p product ( )

= ∩ −1 G G hi,zHzhi,z F, (9.8) DZ 276 9 Homology and Graphs of Pro-C Groups

where F isafreepro-p subgroup of G, and where DZ is the quotient space (whose points are double cosets) DZ = G\H/Hz ={Ghi,zHz | z ∈ Z,i ∈ Iz} z∈Z F = × ∼= . of H/ Z H/ z∈Z H/Hz (see Proposition 5.2.3) under the action of G. (b) If, in addition, the group H and the space Y = {z ∈ Z | Hz = 1} are second- ∈ ∈{ } countable, then one can assume that for each z Z,1 hi,z i∈Iz , i.e., that for each z ∈ Z, G ∩ Hz appears as one of the free factors in the decomposition (9.8). ! = C = Proof We interpret H z∈Z Hz as the fundamental pro- group H C H C H C = Π1( , T (Z)) of the graph of pro- groups ( , T (Z)) over the -tree T(Z) Z . Z . {ω} constructed in Example 6.2.3(b):

Hz1

Hz2 1

1 1 Hz3 1 . 1 . .

In this case the graph of groups (H, T (Z)) is injective, i.e., the canonical morphism ν : H −→ H is injective when restricted to each of the ﬁbers H(t) = Ht (t ∈ T(Z)); furthermore ν(H(t1)) ∩ ν(H(t2)) = 1, for t1 = t2 in T(Z)(see Propo- sition 5.1.6). Hence we may identify H(t) = Ht with its image ν(H(t)) in H . Then the C-standard graph C . S = S H,T(Z) = H/Ht t∈T(Z) of (H, T (Z)) is a C-tree on which H acts continuously (see Corollary 6.3.6). It follows that the G-stabilizer of an element hHt ∈ S is

= ∩ = ∩ −1 ∈ ∈ GhHt G HhHt G hHt h h H,t T(Z) .

In particular, the G-stabilizers of edges of S are trivial, so that Theorem 9.6.1 applies since G is assumed to be second-countable. Note that for the vertices of the form ∈ = hHω S,wealsohaveGhHω 1; furthermore, the subset H/Hω is clopen in S. 9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem 277

Denote by D the space of double cosets . D = G\V(S)= G\H/Hy, y∈Z∪. {ω} with the quotient topology. Then D = DZ ∪. Dω, where . DZ = G\H/Hy and Dω = G\H/Hω. y∈Z

Observe that DZ and Dω are clopen subsets of D. (a) Since G is second-countable, according to Theorem 9.6.1(b) there exists a section . . σ : D = G\V(S)= G\H/Hy −→ V(S)= H/Hy y∈Z∪. {ω} y∈Z∪. {ω} of the projection η : V(S)−→ D = G\V(S) such that the family of G-stabilizers ∈ ∈ ∪. { } Gσ(GhHy ) h H,y Z ω is continuously indexed by the space D = G\V(S), and ( ) ( ) ( ) = = G Gσ(GhHz) Gσ(GhHω) F Gσ(GhHz) F, (9.9) DZ Dω DZ for some free pro-p subgroup F . Next, for each z ∈ Z, deﬁne

{ } = { | ∈ } hi,zHz i∈Iz σ GhHz h H .

Then (9.9) becomes ( )

= ∩ −1 G G hi,zHzhi,z F, Ghi,zHz∈DZ proving part (a). ! ! = = (b) Observe that H z∈Z Hz z∈Y Hz; hence, replacing Z with Y if necessary, we may assume that H and Z are second-countable. Therefore H × Z is second-countable; hence S, and so V(S), is second-countable (see the deﬁnition of S in Sect. 6.3). Since T(Z)is a C-tree, W = 1Hz z ∈ V T(Z) 278 9 Homology and Graphs of Pro-C Groups is a closed subspace of V(S) (see Lemma 6.3.2(d)). Hence, according to Theo- rem 9.6.1(c), there exists a section . . σ : D = G\V(S)= G\H/Hy −→ V(S)= H/Hy y∈Z∪. {ω} y∈Z∪. {ω} of the projection η : V(S)−→ D = G\V(S) such that W ⊆ σ(D). The proof now proceeds as in Part (a), but with the observation that since W ⊆ σ(D), for each x, one of the hi,x may be chosen to be 1.

Corollary 9.6.3 Under the hypotheses of Theorem 9.6.2, assume in addition that G is indecomposable, i.e., G is not the free pro-p product of two nontrivial pro-p = ≤ h ∈ ∈ groups. Then either G Zp or G Hz for some z Z, h H .

Proof Since G is indecomposable, the free pro-p product decomposition of G described in the theorem has only one nontrivial free factor. Therefore, either hi,z hi,z G = G ∩ Hz for some z ∈ Z, in which case G ≤ Hz ,orG = F . In this last case, since G is indecomposable, either G = Zp or G = 1. Chapter 10 The Virtual Cohomological Dimension of Proﬁnite Groups

10.1 Tensor Product of Complexes

We begin with a useful characterization of exactness of a sequence of vector spaces. We assume here that K is a ﬁnite ﬁeld. Let

αn α1 α V : ···−→Vn −→ Vn−1 −→···−→ V0 −→ V −→ 0 be a complex of discrete K-modules (respectively, proﬁnite K-modules) and homomorphisms (respectively, continuous homomorphisms). Recall that being a complex means that the composition of two contiguous mappings is 0. We say that the complex V of discrete K-modules (respectively, proﬁnite K- modules) splits if there exist homomorphisms (respectively, continuous homomorphisms) of K-modules σ : V −→ V0 and σn : Vn −→ Vn+1 such that ασ = idV and + = + = σα α1σ0 idV0 ,σn−1αn αn+1σn idVn (10.1)

(n = 1, 2,...). We say that {σn,σ} is a splitting for the complex V.

Lemma 10.1.1 The complex V of discrete or proﬁnite K-modules is exact if and only if it splits.

Proof Since V is a complex, Im(αn+1) ≤ Ker(αn), for each n ≥ 1 and Im(α1) ≤ Ker(α). Suppose the complex V splits; then α is onto, and if a ∈ Ker(αn) (respectively, a ∈ Ker(α)), using (10.1)wehavea = αn+1(σn(a)) (respectively, a = α1(σ0(a))), so a ∈ Im(αn+1) (respectively, a ∈ Im(α1)); hence V is exact. Con- versely, assume that V is exact. We shall show that V admits a splitting; we do this for the case when V is a sequence of profinite K-modules; for the case of discrete K-modules the proof is similar. Observe that every profinite K-module is projective in the category of profinite K-modules: indeed, such a module is the inverse limit of finite K-modules (see Sect. 1.7), i.e., of finite-dimensional vector spaces over K

© Springer International Publishing AG 2017 279 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_10 280 10 The Virtual Cohomological Dimension of Proﬁnite Groups which certainly are projective. So there is a continuous section σ : V −→ V0 of α; and inductively, assuming the existence of continuous homomorphisms

σ0 : V0 −→ V1,...,σn−1 : Vn−1 −→ Vn satisfying (10.1), use projectivity to construct a continuous homomorphism σn : Vn −→ Vn+1 that makes the diagram

Vn σn ϕ

Vn+1 Im(αn+1) αn+1 commutative, where Id − σ − α if n ≥ 1, ϕ = Vn n 1 n − = IdV0 σα if n 0,

(observe that in fact Im(ϕ) = Im(αn+1), since V is exact). Then clearly the formulas (10.1) are still satisﬁed.

Next we recall the concept of a tensor product of complexes.Let

αn α1 α A : ···−→An −→ An−1 −→···−→ A0 −→ A −→ 0 and

βn β1 β B : ···−→Bn −→ Bn−1 −→···−→ B0 −→ B −→ 0 be complexes of proﬁnite K-modules and continuous homomorphisms. The complete tensor product A ⊗ B = A ⊗ K B of these two complexes is deﬁned to be a sequence

γn γ1 γ C : ···−→Cn −→ Cn−1 −→···−→ C0 −→ C −→ 0 # n where C = A ⊗ B, Cn = = Ai ⊗ Bn−i , γ = α ⊗ β and where γn on the direct i 0 summand Ai ⊗ Bn−i is the map ⊗ + − i ⊗ αi idBn−i ( 1) idAi βn−i, so that "n = ⊗ + − i ⊗ γn αi idBn−i ( 1) idAi βn−i . i=0 Then C is clearly a sequence of proﬁnite K-modules and continuous homomorphisms; one easily checks that in fact it is also a complex (see, for example, Mac Lane 1963, Chap. V, Sect. 9 for more details). 10.2 Tensor Product Induction for a Complex 281

Proposition 10.1.2 If A and B are exact, so is C = A ⊗ B.

{ A A} Proof By Lemma 10.1.1, it suffices to show that C admits a splitting. Let σn ,σ { B B } and σn ,σ be splittings for the exact sequences A and B, respectively. Define A B σ = σ ⊗ σ : C = A ⊗ B −→ C0 = A0 ⊗ B0, and define "n "n+1 σn : Cn = Ai ⊗ Bn−i −→ Cn+1 = Ai ⊗ Bn+1−i i=0 i=0 to be "n = A ⊗ B + A ⊗ ⊕ A ⊗ σn σ α σn σ0 idBn σi idBn−i . i=1

One checks easily that {σn,σ} is a splitting for C.

10.2 Tensor Product Induction for a Complex

Let G be a proﬁnite group and let H be an open subgroup of G;say[G : H ]=s, the index of H in G. Denote by Σ = H\G the set of right cosets of H in G. Then G acts continuously on the right of Σ in a natural way: Σ × G −→ Σ is the map (Hg1,g2) → Hg1g2 (g1,g2 ∈ G). Denote the set Σ by {1,...,s}, where we shall agree that the coset H ∈ H\G is denoted by 1. Then we denote the above action by

(i, g) → ig, (i ∈ Σ,g ∈ G).

Let ρ : G −→ Sym(s) be the corresponding permutation representation: if g ∈ G, ρ(g) is the permutation in Sym(s) which sends j ∈ Σ to jg. Observe that

Ker(ρ) = HG, = −1 where HG x∈G x Hx is the core of H in G, the largest normal subgroup of G contained in H ; since H is open, so is HG. s We endow GΣ = G × ···× G, the group of all functions from Σ to G, with the product topology. Consider a continuous left action of G on GΣ , which we denote by (g, f ) → gf , deﬁned by

gf (j) = f(jg) (g∈ G, j ∈ Σ,f : Σ −→ G).

This action allows us to construct a corresponding wreath product G Sym(s) = GΣ Sym(s), which is obviously a proﬁnite group. We are especially interested in the closed subgroup

H ρ(G)= H Σ ρ(G)≤ G Sym(s). 282 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Choose a right transversal T ={t1 = 1,t2,...,ts} of H in G, where tj is an element of the coset j. Then there exists a well-known group embedding

ϕ : G −→ H˜ = H Σ ρ(G)= H ρ(G) (10.2) of G into the wreath product of H and ρ(G) which is given by

ϕ(g) = fgρ(g), (g ∈ G) where the map fg : Σ −→ H is given by

= −1 ∈ ∈ fg(j) tj gtjg (g G, j Σ). Explicitly, if g ∈ G, then ϕ(g) is the value at g of the composition of the following injective homomorphisms:

δ×ρ innST G δG × ρ(G) G ρ(G) G ρ(G) ,

s where δG : G −→ G × ···×G is the diagonal map, ST : Σ −→ G is the map = ∈ ST (i) ti (i Σ), and innST is the inner automorphism of G ρ(G) associated to the element ST ∈ G ρ(G). (See RZ, Appendix D, for more details using the above notation.) Assume now that A0,A1,... are profinite left [[ FpH ]] -modules, where Fp is the field with p elements. For a natural number n, consider the profinite K-module (complete tensoring ⊗ is assumed to be over K) " G = ⊗ ⊗ tenH (A•)n Ai1 ... Ais . i1+···+is =n

If a ∈ Aj , we write d(a) = j. Next we describe an action of the proﬁnite group ˜ = = Σ G H H Sym(s) H Sym(s) on tenH (A•)n (and therefore, using the embed- G = ∈ ˜ = ∈ Σ ∈ ding (10.2), of G on tenH (A•)n). If x fτ H H Sym(s) (f H ,τ ∈ = Sym(s)) and aj Aij (j 1,...,s), deﬁne ⊗··· ⊗ = − ν ⊗··· ⊗ ∈ ⊗ ⊗ x(a1 as) ( 1) a1 as Ai1τ ... Aisτ , where = = = aj f(j)ajτ (j 1,...,s) and ν d(aj )d(ak). (10.3) jkτ −1

One checks that this defines a continuous multiplication ˜ × G −→ G H tenH (A•)n tenH (A•)n, (10.4) 10.2 Tensor Product Induction for a Complex 283 which we denote by (x, m) → xm. This can be seen by using an inverse limit argument which we briefly sketch: assume first that the Ai are finite; the action of H on each Ai factors through a finite quotient H/U; we may assume that there is an open subgroup U valid for all Ai (0 ≤ i ≤ n); hence (10.4) factors through

× G −→ G (H/U) Sym(s) tenH (A•)n tenH (A•)n, which is obviously continuous since all the groups and modules involved are finite; therefore (10.4) is continuous in this case. For general Ai , express them as inverse limits of finite H -modules (see Sect. 1.7); then, by the previous case, (10.4) can be expressed as an inverse limit of analogous continuous maps obtained from the finite quotients of the Ai ; thus (10.4) is continuous.

Lemma 10.2.1 The multiplication (10.4) deﬁnes a continuous action of H˜ on G tenH (A•)n.

= ∈ G Proof Note that 1m m, for all m tenH (A•)n. Hence it only remains to show = ∈ ˜ ; ∈ G that (x x)m x (xm) (x, x H m tenH (A•)n); and for this it sufﬁces to as- = ⊗··· ⊗ ∈ ⊗ ⊗ sume (using a density argument) that m a1 as Ai1 ... Ais , where ∈ = = = ∈ Σ ∈ aj Aij (j 1,...,s). Say x fτ and x f τ (f, f H ,τ,τ Sym(s)). By deﬁnition of multiplication in the wreath product,

x x = f τ f τ τ ,

where (τ f )(j) = f(jτ ).Sowehave

= − ν ⊗··· ⊗ = − ν+ν ⊗··· ⊗ x (xm) x ( 1) a1 as ( 1) a1 as , where the aj and ν are given by the formulas (10.3), and similarly, = = aj f (j)ajτ (j 1,...,s) and = = ν d aj d ak d(ajτ)d(akτ ). jkτ −1 jτ −1>kτ −1 Therefore,

= = τ aj f (j)f jτ ajτ τ f f (j)ajτ τ . So to complete the proof of the lemma, we just need to verify the signs, i.e., we need to show that

ν + ν ≡ d(aj )d(ak)(mod 2). jkτ −1τ −1 284 10 The Virtual Cohomological Dimension of Proﬁnite Groups

To do this write ν = ν1 + ν2, where ν1 = d(aj )d(ak), ν2 = d(aj )d(ak). jkτ −1 jτ−1>kτ −1 jτ−1τ −1>kτ −1τ −1 jτ−1τ −1

Changing the summation indices, we can rewrite ν as

ν = d(ajτ)d(akτ ) = d(aj )d(ak). jkτ jτ−1τ −1>kτ −1τ −1 = + Write ν ν1 ν2, where = = ν1 d(aj )d(ak), ν2 d(aj )d(ak). jτ−1kτ −1τ −1 jτ−1τ −1>kτ −1τ −1 jk = Now observe that ν2 ν2 (just exchange j and k in ν2). Therefore, + ≡ + + ≡ + = ν ν ν1 ν1 2ν2 ν1 ν1 d(aj )d(ak)(mod 2), jkτ −1τ −1 as desired.

Using the embedding (10.2) we deduce

G Corollary 10.2.2 There exists a continuous action of G on tenH (A•)n such that ⊗··· ⊗ = − ν ⊗··· ⊗ g(a1 as) ( 1) a1 as (10.5) +···+ = ∈ i1 is n, aj Aij , = = aj fg(j)ajg, and ν d(aj )d(ak), jkg−1

= −1 = where fg(j) tj gtjg (j 1,...,s).

G We refer to the action of G on tenH (A•)n considered above as the tensor induced G-action.

Remark 10.2.3 (a) The need for the sign (−1)ν in the deﬁnition of the action (10.3)or(10.5) will become apparent later when we consider Theorem 10.2.8. 10.2 Tensor Product Induction for a Complex 285

(b) Every profinite right H -module A can be made into a profinite left H -module by the usual formula ha = ah−1 (h ∈ H,a ∈ A). Hence the above construction can be carried out in a similar manner if the sequence A ={A0,A1,...} consists of profinite right [[ FpH]] -modules instead. For example, one easily checks that G the action (10.5) translates to the following right action of G on tenH (A•)n:if ∈ ∈ +···+ = g G, aj Aij , i1 is n, then

ν (a ⊗ ... ⊗ a )g = (−1) a ⊗ ... ⊗ a ∈ A − ⊗ ... ⊗ A − , 1 s 1 s i1g 1 isg 1

= −1 = where aj ajg−1 fg(jg ) (j 1,...,s), and where ν = d(aj )d(ak). jkg

G Next we study the structure of the G-modules tenH (A•)n in the following special case. Assume that Xi is a profinite H -space (i = 0, 1,...);letAi =[[FpXi]] be the free profinite Fp-module on the space Xi ; then each Ai is a profinite [[ FpH ]] - module in a natural way (cf. RZ, Sect. 5.7). Starting with these Ai construct the G profinite G-modules tenH (A•)n as above. Observe that

⊗ ⊗ =[[ ]] ⊗ ⊗[[ ]] = ×···× Ai1 ... Ais FpXi1 ... FpXis Fp(Xi1 Xis ) ,

G =[[ ˜ ]] so that tenH (A•)n FpXn is the free profinite Fp-module with basis the profinite space ˜ = . ×···× ; Xn Xi1 Xis i1+···+is =n G furthermore, the action of G on tenH (A•)n is determined by its action on the = ˜ ∪ − ˜ G ∈ ∈ profinite subspace Zn Xn ( Xn) of tenH (A•)n:ifg G, and (x1,...,xs) ×···× Xi1 Xis , then

ν g(x1,...,xs) = (−1) fg(1)x1g,...,fg(s)xsg , (10.6) where ν = d(xj )d(xk), jkg−1 and d(x) = i means that x ∈ Xi .

Proposition 10.2.4 Assume that p is a prime number. Let G be a pro-p group and let H be an open subgroup of G of index s. Let A ={A0 =[[FpX0]] ,A1 = 286 10 The Virtual Cohomological Dimension of Proﬁnite Groups

[[ FpX1]] ,...} be a collection of [[ FpH]] -modules as above. Then the [[ FpG]] - G [[ ]] module tenH (A•)n is FpG -isomorphic to the free profinite Fp-module on the profinite G-space = . ×···× Un Xi1 Xis , i1+···+is =n ∈ ×···× with action of G on Un defined as follows: if (x1,...,xs) Xi1 Xis and g ∈ G, then

g · (x1,...,xs) = fg(1)x1g,...,fg(s)xsg . (10.7)

˜ Proof Of course Xn and Un coincide as topological spaces. Observe that the actions (10.6) and (10.7) differ only by a sign. So if p = 2, they are the same; in this case the result is clear since Zn = Un. G Assume that p is odd. Then we shall show that tenH (A•)n with action (10.6) and [[ FpUn]] with action (10.7)are[[ FpG]] -isomorphic (we already know that they are isomorphic as profinite Fp-modules). To define this isomorphism it suffices to G exhibit a profinite G-subspace Yn of tenH (A•)n which freely generates it as an Fp- ˜ module, and such that Yn is isomorphic to Xn as G-spaces. ˜ ˜ Note that Xn and −Xn are disjoint. We shall construct Yn as a G-subspace of ˜ ˜ ˜ ˜ Zn = Xn ∪ (−Xn). Observe that the map π : Zn = Xn ∪ (−Xn) −→ Un that sends ˜ ±u to u(u∈ Xn) is a map of profinite G-spaces; and, as a map of topological spaces, π is a 2-covering of Un. ˜ We claim that if u ∈ Xn, then gu = −u, for all g ∈ G. Indeed, suppose that gu =−u,forsomeg ∈ G; then the action of g on Σ cannot be trivial (see the 2 value of ν in (10.6)), i.e., g/∈ Ker(ρ) = HG. On the other hand, g u = u means that the procyclic group g acts on {u, gu}, and hence g has a subgroup of index 2, 2 namely g . Therefore, g has even order mod HG. But this contradicts the fact that the finite group G/HG has odd order, since p is odd. Thus the claim follows. ˜ By compactness one deduces that for each u ∈ Xn, there exists a clopen neigh- ˜ bourhood Vu of u in Xn such that GVu ∩ G(−Vu) =∅. Hence π induces a G-isomorphism as profinite G-spaces between GVu and the clopen G-subspace G · π(Vu) of Un. Therefore, for each u ∈ Un there exists a clopen G-invariant : −→ neighbourhood Wu of u in Un and a continuous G-map σWu Wu Zn such = that πσWu idWu . Consider the canonical quotient map η : Un −→ G\Un. Since η is open and continuous, the set {η(Wu) | u ∈ Un} covers G\Un by clopen sets. Observe that −1 Wu = η (η(Wu)). Since G\Un is a profinite space it follows that there exists a finite cover {L1,...,Lt } of G\Un of mutually disjoint clopen subsets such that for −1 each i = 1,...,t there exists a continuous σi : η (Li) −→ Zn such that πσi = −1 −1 −1 { } idη (Li ) and σi is a G-map. Since η (L1),...,η (Lt ) is a cover of Un by mutually disjoint clopen G-subspaces of Un, the union of those σi is a continuous ˜ ˜ section σ : Un −→ Zn of π : Zn = Xn ∪ (−Xn) −→ Un which is a G-map. Define Yn = σ(Un). 10.2 Tensor Product Induction for a Complex 287

Remark 10.2.5 (a) The last part of the argument in the above proof is a consequence of a more general result, which can be proved in an analogous manner: Let π : Z −→ U be a continuous G-map of profinite G-spaces. Assume further that, as a map of topological spaces, π is an n-fold covering, and that for each u ∈ U the stabilizer −1 subgroup Gu of u under the action of G fixes each element of the fiber π (u). Then Z is the disjoint union of n copies of U. (b) In Proposition 10.2.4 one may replace the hypothesis that G is a pro-p group by the assumption that either p = 2 or the index [G : HG] is odd.

Corollary 10.2.6 Assume in addition that the action of H on each Xi is free. Then

(a) the G-stabilizer Gu of any u ∈ Un is ﬁnite; G (b) if G is a torsion-free pro-p group, then each tenH (A•)n is a free proﬁnite [[ FpG]] -module.

= ∈ ×···× ∈ · = Proof (a) Say u (x1,...,xs) Xi1 Xis , and let h Gu. Then h u u implies in particular that fh(1)x1 = x1, since 1h = 1 (recall that under our assump- ∈ tions, 1 Σ is the coset H ). Since the action of H on Xi1 is free, we deduce that = = −1 = = ∩ = 1 fh(1) t1ht1h h,fort1 1. Hence Gu H 1. Since H has finite index, Gu is finite. G =[[ ]] [[ ]] (b) By Proposition 10.2.4, tenH (A•)n FpUn is a FpG -module with G- action on Un defined by (10.7). Since G is torsion-free, it follows from part (a) that G [[ ]] the action of G on Un is free. Therefore, tenH (A•)n is a free FpG -module (cf. RZ, Proposition 5.7.1).

Corollary 10.2.7 If G is assumed to be a torsion-free pro-p group and A0,A1,... [[ ]] G are projective proﬁnite FpH -modules, then each tenH (A•)n is a projective [[ FpG]] -module.

Proof Since Ai is [[ FpH]] -projective, there exists some profinite [[ FpH ]] -module Bi such that Ai ⊕ Bi = Fi is a free profinite [[ FpH ]] -module. Hence Fi can be viewed as a profinite free Fp-module Fi =[[FpXi]] on a profinite space Xi where [[ ]] G H acts freely (cf. RZ, Proposition 5.7.1). Clearly the FpG -module tenH (A•)n is a [[ ]] G direct summand of the FpG -module tenH (F•)n. Since the latter is free according to Corollary 10.2.6, the result follows.

Assume now that

αn α1 α A : ···−→An −→ An−1 −→···−→ A0 −→ Fp −→ 0 is an exact sequence of [[ FpH]] -modules. Consider the s-fold complete tensor product over Fp of A with itself s-times. We shall denote this complex by

G : ···−→ G −→···γn −→γ1 G −→γ −→ tenH (A) tenH (A•)n tenH (A•)0 Fp 0, 288 10 The Virtual Cohomological Dimension of Proﬁnite Groups

s where we have made the identiﬁcation Fp = Fp ⊗ ··· ⊗ Fp. Then

s γ = α ⊗ ··· ⊗ α.

One checks by induction on s that the restriction of γn to the direct summand ⊗ ⊗ +··· = G Ai1 ... Ais (i1 is n)oftenH (A•)n is given by s +···+ (−1)i1 ir−1 id ⊗ ... ⊗ α ⊗ ... ⊗ id , Ai1 ir Ais r=1 where we deﬁne i0 = 0.

Theorem 10.2.8 Let H an open subgroup of index s in a proﬁnite group G. Assume that αn α1 α A : ···−→An −→ An−1 −→···−→ A0 −→ Fp −→ 0 is an exact sequence of proﬁnite left (respectively, right) [[ FpH ]] -modules. Then the s-fold complete tensor product over Fp of A

G : ···−→ G −→···γn −→γ1 G −→γ −→ tenH (A) tenH (A•)n tenH (A•)0 Fp 0, with tensor induced G-action, is an Fp-split exact sequence of proﬁnite [[ FpG]] - modules and continuous [[ FpG]] -homomorphisms. Furthermore, if Ai = 0 for i>d = G = G = = and Ad 0, then tenH (A•)i 0, for i>e, and tenH (A•)e 0, where e sd.

Proof We work with left modules because the formulas are easier to write (see Re- G mark 10.2.3). The last statement follows from the definition of tenH (A). According to Proposition 10.1.2, this sequence is exact, and according to Lemma 10.1.1,itis Fp-split. The map γ is obviously a G-homomorphism, since the action of G on Fp is trivial. So the only remaining task is to show that the Fp-homomorphisms γn are in fact G-maps. ⊗··· ⊗ +···+ To do this we rewrite the definition of the map γn on Ai1 Ais (i1 is = n). First consider auxiliary functions αt,r (1 ≤ t,r ≤ s) defined as follows on ∈ ∞ elements a i=1 Ai : αd(a)(a), if t = r, αt,r(a) = a, if t = r,

= ∈ ⊗··· ⊗ ∈ ⊗··· ⊗ + where d(a) j means that a Aj . Then if a1 as Ai1 Ais (i1 ···+ = ; ∈ = is n aj Aij ,j 1,...,s),wehave s dr γn(a1⊗···⊗as) = (−1) α1,r (a1)⊗···⊗αs,r(as), r=1 where dr = d(a1) +···+d(ar−1),ifr ≥ 2, and d1 = 0. 10.2 Tensor Product Induction for a Complex 289

By linearity and continuity, it sufﬁces to show that

γn g(a1⊗···⊗as) = gγn(a1⊗···⊗as)(g∈ G).

Now,

⊗··· ⊗ = − ν ⊗··· ⊗ γn g(a1 as) γn ( 1) a1 as s + = − ν dr ⊗··· ⊗ ( 1) α1,r a1 αs,r as , r=1

= +···+ where aj and ν are deﬁned in Corollary 10.2.2 and dr d(a1) d(ar−1),if ≥ = ∈ = r 2, d1 0. And on the other hand, since fg(j) H(j 1,...,s)and since each αj,r is an H -map, we have s dr gγn(a1⊗···⊗as) = g (−1) α1,r (a1)⊗···⊗αs,r(as) r=1 s dr νr = (−1) (−1) fg(1)α1g,r(a1g)⊗···⊗fg(s)αsg,r(asg) r=1 s dr νr = (−1) (−1) α1g,r fg(1)a1g ⊗···⊗αsg,r fg(s)asg r=1 s

= − dr − νr ⊗··· ⊗ ( 1) ( 1) α1g,r a1 αsg,r as r=1 s

= − dr − νr ⊗··· ⊗ ( 1) ( 1) α1,rg−1 a1 αs,rg−1 as , r=1 where, if we set δkk = 1 and δjk = 0, if j = k,asusual,wehave νr = d αj,r(aj ) d αk,r(ak) jkg−1 = d(aj ) − δjr d(ak) − δkr jkg−1 = ν − d(aj ) − d(aj ). rjg−1 jg−1>rg−1 290 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Hence, νr + dr = νr + d(aj ) jjg−1 jg−1>rg−1 + d(aj ) + d(aj ) jrg−1 jg−1jg−1 jg−1

= Changing the summation index and taking into account that aj fg(j)ajg, so that = d(aj ) d(ajg),wehave + ≡ + ≡ + ≡ + νr dr ν d(ajg) ν d aj ν drg−1 (mod 2). j

Thus,

s ν+d ⊗··· ⊗ = − rg−1 ⊗··· ⊗ gγn(a1 as) ( 1) α1,rg−1 a1 αs,rg−1 as r=1 s + = − ν dr ⊗··· ⊗ ( 1) α1,r a1 αs,r as r=1

= γn g(a1⊗···⊗as) .

10.3 The Torsion-Free Case

Given a proﬁnite group G and a prime number p,thevirtual cohomological p- dimension of G is deﬁned as vcdp(G) = inf cdp(H ) H ≤o G .

Observe that if G is virtually of finite cohomological p-dimension, then vcdp(G) = cdp(H ), where H is any open subgroup of G of finite cohomological p-dimension; 10.4 Groups Virtually of Finite Cohomological Dimension: Periodicity 291 furthermore, if cdp(G) is finite, then vcdp(G) = cdp(G) (cf. RZ, Theorem 7.3.1; seealsoRemark7.3.2inRZ). On the other hand, if vcdp(G) < ∞ and G is p-torsion-free, we deduce in the next theorem, from the results in Sect. 10.2, that then cdp(G) < ∞.

Theorem 10.3.1 Let p be a prime number. (a) Let G be a proﬁnite group with no elements of order p. If H is an open subgroup of G, then cdp(H ) = cdp(G). (b) Let G be a torsion-free pro-p group. If G contains an open subgroup which is afreepro-p group, then G isafreepro-p group.

Proof Part (b) is a consequence of part (a) since for a pro-p group, freeness is characterized by having cohomological p-dimension at most 1. To prove (a), it sufﬁces to show that if cdp(H ) is ﬁnite, then so is cdp(G) (see Sect. 1.11).

So, assume that cdp(H ) < ∞.LetS be a p-Sylow subgroup of H and let S be a p-Sylow subgroup of G such that S ≤ S . Then cd(S) = cdp(H ) and cd(S ) = cdp(H ) (cf. RZ, Corollary 7.3.3). Hence we may assume that G is a torsion-free pro-p group. Say cd(H ) = d. Then there is an exact sequence

0 −→ Pd −→ · · · −→ P0 −→ Fp −→ 0 where each Pi is a projective proﬁnite [[ FpH]] -module (i.e., this is a projective resolution of Fp of [[ FpH]] -modules) with Pd = 0 (cf. RZ, Proposition 7.1.4). Then according to Theorem 10.2.8 and Corollary 10.2.7, Fp admits a projective resolution of [[ FpG]] -modules of length e = ds, where s is the index of H in G. Thus cd(G) = e<∞.

10.4 Groups Virtually of Finite Cohomological Dimension: Periodicity

Let Fp denote both the ﬁeld with p elements and its underlying additive group. 2 Let G be a proﬁnite group and let it act trivially on the groups Fp and Z/p Z. n n 1 As usual, we write H (G) = H (G, Fp). In this case H (G) = Hom(G, Fp),the group of continuous homomorphisms from G to Fp (see Sect. 1.10). Consider the short exact sequence

2 0 −→ Fp −→ Z/p Z −→ Fp −→ 0 (10.8) of discrete G-modules. Then the Bockstein homomorphism

β : H 1(G) −→ H 2(G) is the usual connecting homomorphism in the long exact sequence associated with (10.8) (see Sect. 1.10). 292 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Proposition 10.4.1 Let G be a proﬁnite group and let f : G −→ Fp be a continuous epimorphism of groups. Put U = Ker(f ) and assume that there exists a natural n number d such that H (U, A) = 0 for a ﬁxed discrete [[ FpG]] -module A and every n ≥ d. Then there exist homomorphisms

ψr : H r (G, A) −→ H r+2(G, A) which are isomorphisms if r>dand an epimorphism for r = d. Explicitly, ψr is the ‘cup product with β(f)’ (cf. RZ, Sect.7.9for the deﬁnition and properties of the cup product):

ψr (h) = β(f)∪ h h ∈ H r (G, A) .

Furthermore, ψr is a morphism of functors H r (G, −) −→ H r+2(G, −), and it commutes with Res (restriction) and Inf (inﬂation).

Proof The last statements follow from the properties of the cup product (cf. RZ, Theorem 7.9.1, Propositions 7.9.4 and 7.9.5). We view f as an element of H 1(G). Put G¯ = G/U . Choose an element x ∈ G¯ ¯ ¯ of order p, so that G = x. Consider the group algebra [FpG], and let ¯ ¯ λ :[FpG]−→[FpG] be multiplication by (1 − x):

¯ λ(m) = (1 − x)m m ∈[FpG] .

The canonical epimorphism G −→ G¯ allows us to consider every G¯ -module as a G-module, and so λ is a homomorphism of [[ FpG]] -modules. Observe that p−1) Ker(λ) = a(1 + x +···+x a ∈ Fp , + p−1 p−1 i Im(λ) = aix ai ∈ Fp, ai = 0 , i=0 i=0 and ¯ Coker(λ) =[FpG]/Im(λ) = a + Im(λ) a ∈ Fp . Note that the actions of G on Ker(λ) and on Coker(λ) are trivial, and so we identify these G-modules with Fp. Therefore we have two short exact sequences

¯ λ 0 −→ Fp −→ [ FpG] −→ Im(λ) −→ 0 (10.9) and ¯ 0 −→ Im(λ) −→ [ FpG]−→Fp −→ 0 (10.10) 10.4 Groups Virtually of Finite Cohomological Dimension: Periodicity 293 of ﬁnite [[ FpG]] -modules. Since A is also an Fp-vector space, the corresponding −⊗ =−⊗ sequences obtained applying the functor A Fp A ¯ 0 −→ A −→ [ FpG]⊗A −→ Im(λ) ⊗ A −→ 0 (10.9 ) and ¯ 0 −→ Im(λ) ⊗ A −→ [ FpG]⊗A −→ A −→ 0 (10.10 ) are exact as well. Hence, applying the properties of the cup product to (10.10) and (10.10 ) and to (10.9) and (10.9 ) we have (cf. RZ, Theorem 7.9.1) a commutative diagram

δ×id δ×id H 0(G)×H r (G, A) H 1(G, Im(λ))×H r (G, A) H 2(G)×H r (G, A)

∪ ∪ ∪

δ δ H r (G, A) H r+1(G, Im(λ) ⊗ A) H r+2(G, A) so that δδ(1) ∪ h = δδ(1 ∪ h) = δδ(h), (10.11) r 0 2 (h ∈ H (G, A), 1 ∈ Fp = H (G)), and hence δδ(1) ∈ H (G).Herewemakethe identiﬁcation Fp ⊗ A = A, and we use the fact that 1 ∪ h = h, which is easily veriﬁed by induction on r. We claim that the map

δδ : H r (G, A) −→ H r+2(G, A) in the second row of the above diagram is an isomorphism if r>dand an epimorphism if r = d. Indeed, since G¯ is ﬁnite, there is an isomorphism of G-modules

[ ¯ ]⊗ =∼ [ ¯ ] = [ ] = G FpG A Hom FpG ,A Hom[FpU] FpG ,A CoindU (A). Here the isomorphism

¯ ¯ [FpG]⊗A → Hom [FpG],A

¯ −1 −1 sends y¯ ⊗ a to fy¯⊗a : G → A, where fy¯⊗a(z)¯ = 0, if z¯ =y ¯ , and fy¯⊗a(y¯ ) = a, ¯ ¯ ¯ for y,¯ z¯ ∈ G, a ∈ A. The actions of G on [FpG]⊗A and Hom([FpG],A) are y(z¯ ⊗ a) = yz ⊗ a and (yf )(z) = f(zy), respectively (y, z ∈ G, a ∈ A). Therefore

r ¯ ∼ r H G, [FpG]⊗A = H (U, A) (r ≥ 0) r ¯ by Shapiro’s Lemma (see Sect. 1.10); and so H (G, [FpG]⊗A) = 0ifr>d. Thus the claim follows from the long exact sequences in cohomology associated with the sequences (10.10 ) and (10.9 ). 294 10 The Virtual Cohomological Dimension of Proﬁnite Groups

We deduce from (10.11) that the homomorphism

+ δδ(1) ∪−:H r (G, A) −→ H r 2(G, A) is an isomorphism if r>cdp(U) and an epimorphism if r = cdp(U). Since the cup product is bilinear, to finish the proof of the proposition it suffices to verify the following assertion: there exists a nonzero element c ∈ Fp such that cβ(f ) = δδ(1). To verify this assertion, assume first that G has order p, so that U ={1}; hence, with our notation, G = x. In this case, since G acts on Fp trivially, one knows that 2 H (G) = Fp (for the cohomology of a finite cyclic group, see, for example, Mac Lane 1995, Chap. IV, Sect. 7); therefore it suffices to show that

β(f) = 0 = δδ(1).

This easily follows by analyzing the long exact sequences in cohomology associated with (10.10) and (10.9): indeed, since

0 0 H G, Im(λ) = Ker(λ) −→ H G, [FpG] = Ker(λ) is an isomorphism, the connecting homomorphism δ : H 0(G) → H 1(G, Im(λ)) 1 arising from (10.10) is an injection. And since H (G, [FpG]) = 0, the connecting homomorphism

δ : H 1 G, Im(λ) → H 2(G) arising from (10.9) is also an injection. Hence δδ(1) = 0. Now, one easily sees that H 1(G) −→ H 1(G, Z/p2Z) is an isomorphism, so that the Bockstein homomorphism β : H 1(G) −→ H 2(G) is an injection in this case; hence β(f) = 0. There- fore, when G has order p we have veriﬁed the assertion. For a general group G the assertion follows from the above case and the commutativity of the diagram

δ δ β H 0(G) H 1(G, Im(λ)) H 2(G) H 1(G)

δ δ H 0(G/U) H 1(G/U, Im(λ)) H 2(G/U) H 1(G/U). β

As a consequence we have the following periodicity result for certain groups of ﬁnite virtual cohomological dimension.

Corollary 10.4.2 Let G be a proﬁnite group and let 0 = f ∈ H 1(G). Put U = Ker(f ), and assume that cdp(U) = d is ﬁnite. Then for every discrete [[ FpG]] - module A, ‘cup product with β(f)’ is an isomorphism H r (G, A) =∼ H r+2(G, A), if r>d, and an epimorphism H d (G, A) −→ H d+2(G, A). 10.5 The Torsion Case 295

We ﬁnish this section with a proposition on ﬁnite groups which uses Proposi- tion 10.4.1, among other results.

Proposition 10.4.3 Let G be a ﬁnite group and let p be a prime number.

(a) Let A be a left [FpG]-module. Assume that for each subgroup T of order p of G we have H 1(T , A) = 0. Then H n(G, A) = 0 for every n ≥ 1. Dually, (b) Let B be a right [FpG]-module. Assume that for each subgroup T of G of order p we have H1(T , B) = 0. Then Hn(G, B) = 0 for every n ≥ 1.

Proof We prove (a). Let P be a p-Sylow subgroup of G. Since pA = 0, we have pH n(G, A) = 0 = pH n(H, A). So the corestriction map

H n(P, A) −→ H n(G, A) is a surjection for n ≥ 1 (cf. RZ, Corollary 6.7.7). Thus we may assume that G is a p-group. If G is elementary abelian (a direct product of groups of order p), then the result follows from the Künneth formula (cf. Brown 1982, Corollary 5.8). So we assume that G is not elementary abelian. For ﬁnite p-groups we know that if H j (G, A) = 0, for some j ≥ 1, then H n(G, A) = 0 for all n ≥ 1 (cf. Serre 1968, Théorème IX.5). Proceeding inductively we may assume that H n(U, A) = 0 for all proper subgroups U of G and all n ≥ 1. Since G is not elementary abelian, there exist ele- 1 2r ments z1,...,zr ∈ H (G) such that the cup product β(z1) ∪···∪β(zr ) ∈ H (G) is zero (cf. Serre 1965, Proposition 4). So, iterating Proposition 10.4.1, we get H 2(G, A) = 0, and thus H n(G, A) = 0 for every n ≥ 1.

10.5 The Torsion Case

This section is dedicated to proving a more general result than Theorem 10.3.1;it describes the higher cohomology groups of G when vcdp(G) < ∞ and G contains subgroups of order p.

Lemma 10.5.1 Let p be a prime number. Assume that the proﬁnite group G contains an open subgroup H with no subgroups of order p (e.g., if cdp(H ) < ∞). Let T be the set of all subgroups of G of order p. Then, (a) T has in a natural way the structure of a proﬁnite space; (b) if G is second-countable, so is T .

Proof Let U be a fundamental system of neighbourhoods of 1 consisting of open normal subgroups U of G contained in H . For each U ∈ U,letT (U) denote the set of all subgroups of G/U of order p.IfP is a subgroup of G of order p, then 296 10 The Virtual Cohomological Dimension of Proﬁnite Groups

P ∩ H = 1; hence the image of P in G/U is in T (U). Therefore T = lim T (U). ←− This proves (a). To prove (b) observe that if G is second-countable, we may choose U ={U1,U2,...} so that U1 ≥ U2 ≥···. Then T = T lim←− (Ui) i∈N is a second-countable proﬁnite space (see Sect. 1.2).

Observe that T is a right G-space by means of the natural continuous action on the right T × G −→ T given by conjugation: (T , g) → T g = g−1Tg (T ∈ T ,g∈ G). If A is a discrete left [[ FpG]] -module, then the group of all continuous maps from T to A,C(T ,A)= Hom([[ FpT ]] ,A) becomes a discrete left [[ FpG]] -module (one can see an explicit proof of this assertion in RZ, Sect. 6.10) with action deﬁned as follows: if f ∈ C(T ,A)and x ∈ G, then (xf )(T ) = xf (T x). Then there is a natural injective homomorphism of [[ FpG]] -modules

ε : A −→ C(T ,A) that sends a ∈ A to the constant map εa : T −→ A with value a. Dually (cf. RZ, Proposition 6.3.6), if B is a proﬁnite right [[ FpG]] -module, the complete tensor product B ⊗[[FpT ]] is a proﬁnite right [[ FpG]] -module with action

(b ⊗ T)x= bx ⊗ T x (x ∈ G, b ∈ B,T ∈ T ).

Then there is a natural continuous [[ FpG]] -epimorphism ε : B ⊗[[FpT ]] −→ A which sends b ⊗ t to b (b ∈ B,t ∈ T ).

Theorem 10.5.2 Let G be a proﬁnite group without subgroups isomorphic to Cp × Cp (Cp denotes the multiplicative group of order p). Assume that H is an open subgroup of G of ﬁnite cohomological p-dimension d, i.e., vcdp(G) = d.

(a) Let A be a discrete left [[ FpG]] -module. Then the natural homomorphism

ε : A −→ C(T ,A)= Hom [[ FpT ]] ,A

induces isomorphisms

εn : H n(G, A) −→ H n G, C(T ,A)

for every n>d. 10.5 The Torsion Case 297

Or, dually, (b) If B is a proﬁnite right [[ FpG]] -module, the epimorphism ε : B ⊗[[FpT ]] −→ B

induces isomorphisms

εn : Hn G, B ⊗[[FpT ]] −→ Hn(G, B)

for each n>d.

Remark 10.5.3 Theorem 10.3.1 is formally a special case of Theorem 10.5.2.In- deed, assume that cdp(H ) = d is ﬁnite. Observe that in this case T =∅, and so [[ FpT ]] = 0. It follows from Theorem 10.5.2 that

n n H (G, Fp) = H G, Hom [[ FpT ]] , Fp = 0 if n>d. Therefore cdp(G) = d (cf. RZ, Proposition 7.1.4). However, it is more natural (and easier) to prove Theorem 10.3.1 directly, as we have done here, and then consider only the case when G has p-torsion.

The proof of Theorem 10.5.2 occupies several pages, and we shall present it using some intermediate results (the conclusion of the proof appears after Proposi- tion 10.5.8). We start with a few comments. To see that (a) and (b) in the above theorem are dual statements, put A = Hom(B, Fp). Then observe (cf. RZ, Propositions 6.3.6 and 5.5.4) that

∼ ∼ Hom B ⊗[[FpT ]] , Fp = Hom [[ FpT ]] , Hom(B, Fp) = C(T ,A),

n and the dual of Hn(G, B) is H (G, A). If T =∅, Theorem 10.5.2 is a consequence of Theorem 10.3.1. Therefore, from now on we shall assume that T = ∅, i.e., that G contains elements of order p. In Theorem 10.5.2 it sufﬁces to prove either (a) or (b). In fact, during the proof of this theorem we shall frequently switch from cohomology to homology and vice versa, depending on what is more convenient on each occasion. As we shall see, one of the crucial steps in the proof of Theorem 10.5.2 is the following proposition.

Proposition 10.5.4 Let G be a profinite group and let H be an open subgroup of G. Assume that H has finite cohomological p-dimension cdp(H ) = d. Then there exists an integer e ≥ 0 with the following property: 1 (a) whenever A is a discrete left [[ FpG]] -module such that H (T , A) = 0 for every subgroup T of G of order p, one has H n(G, A) = 0, for all n>e. Or, dually, (b) whenever B is a profinite right [[ FpG]] -module such that H1(T , B) = 0 for every subgroup T of G of order p, one has Hn(G, B) = 0, for all n>e. 298 10 The Virtual Cohomological Dimension of Profinite Groups

Proof We prove part (a). Replacing H by its core in G, we may assume that H is normal in G (cf. RZ, Theorem 7.3.1). Let S be a p-Sylow subgroup of G, and observe that pHn(S, A) = 0, because pA = 0. Since H ∩ S is a p-Sylow subgroup of H ,wehavecdp(H ∩ S) = cdp(H ) = d. The restriction map

Res : H n(G, A) −→ H n(S, A) is an injection (cf. RZ, Corollary 6.7.7). So it sufﬁces to show that H n(S, A) = 0, and thus, replacing G with S and H with H ∩ S, we may assume that G is a pro-p group. The initial step in the proof of this proposition is the construction of a spectral sequence that converges to H •(G, A) (for facts about spectral sequences and the notation we use here, the reader may consult RZ, Appendix A). Since cdp(H ) = d, there exists a projective resolution of proﬁnite [[ FpH ]] -modules

R : 0 −→ Rd −→···−→R1 −→ R0 −→ Fp −→ 0 of Fp of length d (cf. RZ, Proposition 7.1.4). Set s =[G : H ], and construct the G s-fold tensor power tenH (R) of the complex R with tensor induced G-action. Then the complex

0 −→ Pe −→ Pe−1 −→···−→P1 −→ P0 −→ Fp −→ 0 (10.12) = G = = (where we have put Pi tenH (R•)i (i 0, 1,...)) is exact, and its length is e ds; moreover, (10.12)isFp-split (see Theorem 10.2.8). Hence, applying the functor − = − [[ ]] Hom( ,A) HomFp ( ,A) one obtains an exact sequence of discrete FpG - modules

0 → A → Hom(P0,A)→ Hom(P1,A)→···→Hom(Pe,A)→ 0. (10.13)

x −1 (The action of G on each Hom(Pi,A) is f(m)= xf (x m) (x ∈ G, m ∈ Pi).) Deﬁne a double complex K = (Kr,t ) Cr (G, Hom(P , A)), if r, t ≥ 0, Kr,t = t 0, otherwise,

r where C (G, Hom(Pt ,A)) is the discrete group of all continuous mappings f : r+1 G −→ Hom(Pt ,A)such that

f(xx0,...,xxr ) = xf (x0,...,xr )(x,xi ∈ G)

(see Sect. 1.10). Consider the two spectral sequences associated with this double complex. The exactness of the functor Cr (G, −) (cf. RZ, Lemma 6.5.4) and the exactness of the sequence (10.13) imply that the ﬁrst spectral sequence collapses; indeed,

r,t = t r = E1 H C G, Hom(P•,A) 0, 10.5 The Torsion Case 299 if t>0. Furthermore, r,0 = r ≥ E1 C (G, A) (r 0), and so r,0 = r ≥ E2 H (G, A) (r 0). r,0 = r,0 On the other hand, E2 E∞ (cf. RZ, Propositions A2.5 and A2.1). By Theo- remA4.1inRZ, E ⇒ H •(Tot(K)) and

r,0 r r r−1 r E∞ = F H Tot(K) = F H Tot(K) =···

= F 0H r Tot(K) = H r Tot(K) .

Therefore E ⇒ H •(G, A), where H •(G, A) has the trivial ﬁltration. Moreover, according to Theorem A4.1 in RZ, the second spectral sequence E of K converges to H •(G, A) as well. We know (cf. RZ, Sect. A4) that

r,t = t •,r = t • = t E1 H K H C G, Hom(Pr ,A) H G, Hom(Pr ,A) . We now use this second spectral sequence to prove the proposition. We must show n = r,t = that H (G, A) 0ifn>e. For this it sufﬁces to prove that E1 0ift>0: r,t indeed, if this is the case, one has E∞ = 0 whenever r + t>e, since Pr = 0for r>e; therefore E ⇒ H •(G, A) (cf. RZ, Theorem A4.1), implies that

H n(G, A) = F 0 H n(G, A) = F 1 H n(G, A) =···= F n H n(G, A) = 0 if n>e(cf. RZ, Proposition A2.1). Thus to ﬁnish the proof we need to establish the following

t Claim H (G, Hom(Pr ,A))= 0, if t>0.

We shall in fact prove the dual statement: Ht (G, B ⊗ Pr ) = 0, for t>0 and for each profinite right [[ FpG]] -module B such that H1(T , B) = 0 whenever T is a subgroup of order p of G. [[ ]] = G Since each Ri is a projective profinite FpH -module, Pr tenH (R•)r is a di- G [[ ]] rect summand of tenH (F•)r where each Fi is free as an FpH -module and so it has the form Fi =[[FpXi]] with Xi a profinite space where H acts freely (see the argument in the proof of Corollary 10.2.7). Since Ht (G, B ⊗ Pr ) is a direct summand of ⊗ G Ht (G, B tenH (F•)r ), we may replace P• by F•, and so, using Proposition 10.2.4 we are reduced to showing that

Ht G, B ⊗[[FpUr ]] = 0, for t>0, 300 10 The Virtual Cohomological Dimension of Profinite Groups where G acts on Ur according to the (equivalent for the right action of) formula (10.7). Furthermore, according to Corollary 10.2.6, the stabilizer Gu is finite for every u ∈ Ur . Let η : Ur −→ Ur /G be the canonical epimorphism. By Proposition 5.5.4 there exists an isomorphism of profinite abelian groups " " ∼ −1 ∼ [[ FpUr ]] = Fpη (u)¯ = Fp(uG) , u¯∈Ur /G u¯∈Ur /G where for u ∈ Ur , we use the notation u¯ = η(u) ∈ Ur /G. Notice that this is also an isomorphism of [[ FpG]] -modules since each [[ Fp(uG)]] is in fact an [[ FpG]] - module. Since complete tensoring commutes with direct sums indexed by a profinite space (see Corollary 9.1.2), " B ⊗[[FpUr ]] = B ⊗ Fp(uG) . u¯∈Ur /G

Hence, by Theorem 9.1.3, " Ht G, B ⊗[[FpUr ]] = Ht G, B ⊗ Fp(uG) . u¯∈Ur /G

Thus it sufﬁces to prove that if u ∈ Ur , then

Ht G, B ⊗ Fp(uG) = 0, for t>0.

We assert that

Ht G, B ⊗ Fp(uG) = Ht (Gu,B), where Gu is the G-stabilizer of u. Indeed, since uG and Gu\G are isomorphic as [[ \ ]] =∼ ⊗ [[ ]] proﬁnite G-spaces, and since Fp(Gu G) Fp [[ FpGu]] FpG , the assertion is a consequence of Shapiro’s Lemma (see Sect. 1.10). Hence we need to prove that Ht (Gu,B)= 0, for t>0, if H1(T , B) = 0 for all subgroups T of order p of Gu. By Corollary 10.2.6, Gu is ﬁnite. So the result is a consequence of Proposi- tion 10.4.3.

Let T = ∅ and let A be a discrete left [[ FpG]] -module. Consider the [[ FpG]] - homomorphism ε : A −→ C(T ,A)deﬁned above. Denote its cokernel by P(A),so that one has a short exact sequence

ε 0 −→ A −→ C(T ,A)−→ P(A)−→ 0. (10.14)

Since C(T , −) = Hom([[ FpT ]] , −) is an additive exact functor, so is P(−). Rather than proving Theorem 10.5.2 as stated, we shall ﬁnd it more convenient to use the following equivalent form. 10.5 The Torsion Case 301

Lemma 10.5.5 Assume T = ∅. Then, under the hypotheses of Theorem 10.5.2, its conclusion is equivalent to

H n G, P (A) = 0 for n>dand every discrete left [[ FpG]] -module A.

Proof Consider the long exact sequence

···−→H d+1(G, A) −→ H d+1 G, C(T ,A) −→ H d+1 G, P (A) −→··· associated with the short exact sequence (10.14). Then if

H n(G, A) → H n G, C(T ,A) is an isomorphism for n>d, it follows that H n(G, P (A)) = 0forn>d. Conversely, assume that H n(G, P (A)) = 0forn>d and every discrete left [[ FpG]] -module A. Then from the above long exact sequence, we deduce that the homomorphism

H n(G, A) → H n G, C(T ,A) is an isomorphism for n ≥ d + 2, and an epimorphism for n = d + 1. Therefore it remains to show that ρ : H n(G, A) → H n(G, C(T ,A)) is an injection for n = d + 1. In order to do this, form a commutative diagram of [[ FpG]] -modules and [[ FpG]] -homomorphisms

→ → G →π → 0 L CoindH (A) A 0 ↓ ε ↓ ε ↓ ε → T → T G → T → 0 C( ,L) C( , CoindH (A)) C( ,A) 0

G where CoindH (A) is the coinduced module associated with A (as an H -module; see [[ ]] ∈ G Sect. 1.10); π is the FpG -homomorphism deﬁned as follows: if f CoindH (A), put = −1 π(f) ti f(ti), i where {ti}i is a right transversal of H in G (π is sometimes called the trace map); and where L = Ker(π). The lower row of the diagram is obtained from the ﬁrst by applying the exact functor C(T , −), and so it is exact as well. Correspondingly one gets a commutative diagram in cohomology (cf. RZ, Proposition 6.2.2)

n G → n →α n+1 H (G, CoindH (A)) H (G, A) H (G, L) ↓↓ρ ↓¯ε n T G → n T → n+1 T H (G, C( , CoindH (A))) H (G, C( ,A)) H (G, C( ,L)) with exact rows. Using Shapiro’s Lemma (see Sect. 1.10) and the fact that = n G = n = ≥ + cdp(H ) 0, we deduce that H (G, CoindH (A)) H (H, A) 0forn d 1. 302 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Hence α is an isomorphism for n ≥ d + 1. Finally, observe that

ε¯ : H n+1(G, L) −→ H n+1 G, C(T ,L) is an isomorphism if n = d + 1 as shown above. Therefore

ρ : H d+1(G, A) −→ H d+1 G, C(T ,A) is an injection, as desired.

Lemma 10.5.6 Let G be a proﬁnite group and let H be an open subgroup of G of ﬁnite cohomological p-dimension d.

(a) Let X be a proﬁnite right G-space. If A is a discrete left [[ FpG]] -module, then

n G = H G, C X, CoindH (A) 0

for n>d. Or, dually, (b) Let X be a proﬁnite left G-space. If B is a proﬁnite right [[ FpG]] -module, then

G ⊗[[ ]] = Hn G, IndH (B) FpX 0

for n>d.

Proof We prove part (b). Expressing X as an inverse limit of finite G-spaces, and observing that complete tensoring and homology commute with inverse limits (cf. RZ, Lemma 5.6.4, Lemma 5.5.2 and Proposition 6.5.7), we may assume that X is finite. Write X as a finite disjoint union of its G-orbits. Since complete tensoring and homology commute with direct sums, we may assume that G acts transitively on X, and so X = K\G, for some open subgroup K of G. Then, using Shapiro’s Lemma (see Sect. 1.10)

G ⊗[[ ]] = G ⊗ [[ \ ]] Hn G, IndH (B) FpX Hn G, IndH (B) Fp K G

= G ⊗ [[ ]] Hn G, IndH (B) [[ FpK]] FpG

= G G Hn G, IndK IndH (B)

= G G Hn K,resK IndH (B) ,

G where resH indicates simply restriction of scalars from G to K. Next (cf. RZ, Propo- sition 6.11.2)

" − G G =∼ K e 1He resK IndH (B) IndK∩e−1HeresK∩e−1He(Be), e∈E 10.5 The Torsion Case 303 where E is a set of representatives of the ﬁnite set of double cosets H \G/K. Hence we are reduced to proving that

K e−1He = Hn K,IndK∩e−1HeresK∩eHe−1 (Be) 0 for all n>d and all e ∈ G. Therefore, using Shapiro’s Lemma again, we see that it sufﬁces to show that

−1 Hn K ∩ e He,B = 0, −1 for all e ∈ G,alln>d and all proﬁnite right [[ Fp(K ∩ eHe )]] -modules B .But this is clear since

−1 −1 cdp K ∩ e He ≤ cdp e He = cdp(H ) = d

(see Sect. 1.11).

Lemma 10.5.7 Under the hypotheses of Theorem 10.5.2, let A be a discrete G- module, and let f : T −→ A be a continuous map. Fix T ∈ T , and choose a generator t of T . Assume f(T)= 0. Then there exist disjoint clopen subsets W and W of T such that (a) W is a T -invariant clopen neighbourhood of T in T ; (b) f vanishes on W ; (c) T = W ∪. W ∪. t−1W t ∪. ···∪. t−(p−1)W tp−1.

T Proof Since A is discrete and is profinite, we may choose a clopen neighbour- = = p−1 −i i = hood V of T such that f(V) 0. Set W i=0 t Vt . Then obviously f(W) 0 and t−iWti = W for all i. Note that T acts freely on the profinite space T − W , since G does not contain any subgroup of the form Cp × Cp. Therefore there exists a continuous section (T − W)/T −→ T − W (see Sect. 1.3). Since T is finite, the image W of this section is clopen in T − W , and so in T . Clearly W and W satisfy (a), (b) and (c).

Proposition 10.5.8 Let G and H be as in Theorem 10.5.2. Let T ∈ T . Then the map

εn : H n(T , A) −→ H n T,C(T ,A) induced by the canonical homomorphism ε : A −→ C(T ,A)is an isomorphism for every n ≥ 1 and every discrete [[ZG]] -module A.

Proof First we claim that it is enough to prove this proposition for n = 1. To verify this claim we consider the dual statement:

εn : Hn T,B⊗[[ZT ]] −→ Hn(T , B) 304 10 The Virtual Cohomological Dimension of Profinite Groups is an isomorphism for each n ≥ 1 and every profinite right [[ ZG]] -module B (εn is induced by the canonical homomorphism ε : B ⊗[[ ZT ]] −→ B). This claim follows because {H1(T , −), H2(T , −), . . . } is a universal sequence of functors (see Sect. 1.10); but we show the claim explicitly by a dimension shifting argument. For a profinite right [[ZG]] -module B,let

0 −→ B −→ P −→ B −→ 0 be an exact sequence of right proﬁnite [[ZG]] -modules with P projective. Then there is a commutative diagram

0 → B ⊗[[ ZT ]] −→ P ⊗[[ ZT ]] −→ B ⊗[[ ZT ]] → 0 ↓↓↓ 0 → B −→ P −→ B → 0 with exact rows. Correspondingly we have a commutative diagram

δ Hn+1(T , P ⊗[[ZT ]] ) → Hn+1(T , B ⊗[[ZT ]] ) → Hn(T , B ⊗[[ZT ]] ) ↓↓↓ δ Hn+1(T , P ) → Hn+1(T , B) → Hn(T , B ) with exact rows (cf. RZ, Propositions 5.5.3 and 6.3.4). Since P and P ⊗[[ ZT ]] are projective [[ ZT ]] -modules, Hn(T , P ) = Hn(T , P ⊗[[ZT ]] ) = 0, for n ≥ 1; so we obtain that the maps δ in this diagram are isomorphisms if n ≥ 1. Hence the claim follows by induction. Therefore we must prove that

ε1 : H 1(T , A) −→ H 1 T,C(T ,A) is an isomorphism. Observe that ε : A −→ C(T ,A) splits as a T -map; indeed, de- ﬁne σ : C(T ,A)−→ A by σ(f)= f(T); then σ is a T -map and clearly σε= idA. Denote by σ 1 : H 1(T , C(T ,A))−→ H 1(T , A) the homomorphism induced by σ . Then σ 1ε1 = id, and hence ε1 is an injection. It remains to show that ε1 is an epimorphism. Recall that an element of H 1(T , C(T ,A))is represented by a derivation d : T −→ C(T ,A)(see Sect. 1.10). Let t be a generator of T . Then d is completely determined by its value on t;say d(t) = f ∈ C(T ,A). From the deﬁnition of derivation one sees that

i−1 i−1 d ti = (1+···+t )d(t) = (1+···+t )f

(recall that we are denoting the action of an element g ∈ G on an element ε ∈ T g g = g ∈ T = +···+ p−1 C( ,A)by ε; and ε(T1) gε(T1 ), T1 ). Set N 1 t . Then

Nf = Nd(t) = d tp = d(1) = 0. 10.5 The Torsion Case 305

In particular, (Nf )(T ) = Nf (T ) = 0. Consider the derivation

d : T −→ A such that d (t) = f(T). Denote by d¯ ∈ H 1(T , C(T ,A)) and d¯ ∈ H 1(T , A) the elements of the cohomology groups represented by d and d , respectively. We assert that ε1(d¯ ) = d¯. To see this we show that there exists an inner derivation

ι : T −→ C(T ,A) such that ι = d − εd . It sufﬁces to check this at t, i.e., we must show the existence of an inner derivation ι such that

ι(t) = d(t)− εd (t) = f − εf(T), where εf(T) : T −→ A is the constant function with value f(T). Now, such an inner derivation would be deﬁned by some h ∈ C(T ,A)so that

i ι ti = h − t h, ∀i.

Since a derivation of T is determined by its value at t, we need a continuous function (1−t) h : T −→ A such that h = f − εf(T). Renaming f − εf(T) and calling it f again, we have f(T)= 0. Therefore, we have reduced the problem to the following: given f ∈ C(T ,A)such that Nf = 0 and f(T)= 0, we need to prove the existence of h ∈ C(T ,A)such that − (1 t)h = f. (10.15)

Let W and W be as in Lemma 10.5.7. Deﬁne h as follows: for S ∈ W ∪ W , h(S) = 0, and if S ∈ W & ' i i −j h t−iSti =− t f t−iSti =− t−j f tj−iSti−j j=1 j=1 for i = 1,...,p− 1. To verify (10.15) we must prove that [(1−t)h](L) = f(L)for all L ∈ T .IfL ∈ W , then [(1−t)h](L) = h(L) − th(t−1Lt) = 0 = f(L), since W is T -invariant. If L ∈ T − W , then L = t−iSti for some S ∈ W , i = 0, 1,...,p− 1. We distinguish three cases.

Case 1. i = 0. Then

(1−t)h (S) = h(S) − th t−1St =−th t−1St = t t−1f(S) = f(S). 306 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Case 2. 1 ≤ i ≤ p − 2. Then

(1−t)h t−iSti = h t−iSti − th t−(i+1)Stt+1 i i+1 =− t−j f tj−iSti−j + t t−j f tj−i−1Sti+1−j j=1 j=1

= f t−iSti .

Case 3. i = p − 1. Then

(1−t)h t−(p−1)Stp−1

= h t−(p−1)Stp−1 − th(S)= h t−(p−1)Stp−1

=−t−1f t−(p−2)Stp−2 − t−2f t−(p−3)Stp−3 −···−t−(p−1)f(S)

−1 −2 −(p−1) =− (1+t +t +···+t )f t−(p−1)Stp−1 + f t−(p−1)Stp−1

=− Nf t−(p−1)Stp−1 + f t−(p−1)Stp−1 = f t−(p−1)Stp−1 , since Nf = 0.

End of the proof of Theorem 10.5.2 We shall prove part (a). As pointed out earlier we may assume that the group G has no elements of order p, i.e., T = ∅. According to Lemma 10.5.5, in this case it sufﬁces to show that

H n G, P (A) = 0, for n>dand every discrete left [[ FpG]] -module A, where

P(−) = C(T , −)/ε(−) is the functor deﬁned in (10.14). To verify this we claim that it sufﬁces to prove the following two assertions: (i) there exists a natural number e such that H n(G, P (A)) = 0, for n>eand every discrete [[ FpG]] -module A; and n G = [[ ]] (ii) H (G, P (CoindH (A))) 0, for n>dand any FpG -module A. Indeed, assume that (i) and (ii) hold. As in the proof of Lemma 10.5.6, consider the short exact sequence

−→ −→ G −→π −→ 0 L CoindH (A) A 0.

Applying the exact functor P(−), we get an exact sequence

−→ −→ G −→ −→ 0 P(L) P CoindH (A) P(A) 0, 10.5 The Torsion Case 307 and correspondingly a long exact sequence in cohomology

···→ n G → n → n+1 →··· H G, P CoindH (A) H G, P (A) H G, P (L) .

Therefore, from (ii) we deduce that H n(G, P (A)) =∼ H n+1(G, P (L)),ifn>d; and so, using (i), we deduce that H n(G, P (A)) = 0, if n>d, proving the claim. Hence, the only remaining task is to prove (i) and (ii). Let T ∈ T . From the short exact sequence (10.14) we get an exact sequence of cohomology groups

···→H n(T , A) → H n T,C(T ,A) → H n T,P(A) → H n+1(T , A) →··· .

Since H n(T , A) −→ H n(T , C(T ,A)) is an isomorphism if n ≥ 1 (see Propo- sition 10.5.8), we infer that H n(T , P (A)) = 0ifn ≥ 1. Therefore by Proposi- tion 10.5.4, H n(G, P (A)) = 0, if n>e, where e is a certain ﬁxed natural number. This proves (i). G To show (ii), consider the sequence (10.14), after replacing A with CoindH (A):

→ G −→ε T G −→ G −→ 0 CoindH (A) C , CoindH (A) P CoindH (A) 0 Form the corresponding long exact sequence in cohomology

···−→ n T G −→ n G H G, C , CoindH (A) H G, P CoindH (A)

−→ n+1 G −→··· H G, CoindH (A) .

Using Shapiro’s Lemma (see Sect. 1.10) and the hypothesis cdp(H ) = d, we deduce that

n+1 G = n+1 = H G, CoindH (A) H (H, A) 0, − n T G = if n>d 1. On the other hand, H (G, C( , CoindH (A))) 0, if n>d,by n G = Lemma 10.5.6. Therefore, H (G, P (CoindH (A))) 0, if n>d, proving (ii).

Since the action of G on the proﬁnite space T of subgroups of G of order p is continuous, the quotient space T /G consisting of the conjugacy classes of subgroups of order p is also proﬁnite. Let

μ : T −→ T /G be the canonical quotient map. Then it follows from Proposition 5.5.4 that the free Fp-module [[ FpT ]] on the space T can be expressed as a direct sum " −1 [[ FpT ]] = Fp μ (τ) τ∈T /G

−1 of free Fp-modules [[ Fp(μ (τ))]] continuously indexed by the proﬁnite space T /G. Since complete tensoring commutes with such direct sums (see Corol- lary 9.1.2), part (a) of the following proposition is a consequence of Theorems 9.1.1 and 10.5.2. 308 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Proposition 10.5.9 Assume the same hypotheses as in Theorem 10.5.2. Then (a) The epimorphism B ⊗[[FpT ]] −→ B induces continuous homomorphisms " −1 Hn G, B ⊗ Fp μ (τ) −→ Hn(G, B) τ∈T /G

which are isomorphisms for n>d= cdp(H ). (b) Let T ∈ T , and let τ = μ(T ). Then

−1 ∼ Hn G, B ⊗ Fp μ (τ) = Hn NG(T ), B ,

where NG(T ) denotes the normalizer of T in G.

Proof Only the proof of part (b) remains. Note that the space μ−1(τ) is the con- −1 jugacy class of the subgroup T of G. Hence the spaces μ (τ) and NG(T )\G are homeomorphic as G-spaces. Therefore the result follows from Lemma 9.2.2(b).

Corollary 10.5.10 The same hypotheses as in Theorem 10.5.2. Assume in addition that the quotient map μ : T −→ T /G admits a continuous section σ : T /G −→ T . Then the family of normalizer subgroups NG σ(τ) τ ∈ T /G of G is continuously indexed by the proﬁnite space T /G, and if n>d= cdp(H ), then the corestriction map (see Proposition 9.2.3) " Hn NG σ(τ) ,B −→ Hn(G, B) τ∈T /G is an isomorphism.

Proof The family of closed subgroups {NG(T ) | T ∈ T } is continuously indexed by the space T by Lemma 5.2.2.So NG σ(τ) τ ∈ T /G = NG(T ) T ∈ Im(σ ) is continuously indexed by the space T /G, since Im(σ ) is a closed subset of T .By Proposition 9.2.3 there is deﬁned a corestriction map " Hn NG σ(τ) ,B −→ Hn(G, B) τ∈T /G which can be identiﬁed with the map " −1 Hn G, B ⊗ Fp μ (τ) −→ Hn(G, B), τ∈T /G according to Lemma 9.2.2(c). Since the latter map is an isomorphism for n>d= cdp(H ) by Proposition 10.5.9, the result follows. 10.6 Pro-p Groups with a Free Subgroup of Index p 309

10.6 Pro-p Groups with a Free Subgroup of Index p

Consider a pro-p group G that contains a closed subgroup F of index p which is free pro-p. As before, T denotes the proﬁnite G-space of all subgroups T of G of order p. Observe that F G;so,ifT ∈ T , then G = F T . We describe in Theorem 10.6.3 the structure of G as a free pro-p product under some additional conditions. If T =∅, then G is a free pro-p group according to Theorem 10.3.1. So, from now on in this section we shall assume that G has torsion, i.e., T = ∅. Choose a continuous epimorphism

fG : G −→ Fp such that Ker(fG) = F . By the dual of Proposition 10.4.1, there exist homomorphisms ψr = ψr (B) : Hr+2(G, B) −→ Hr (G, B) which are isomorphisms for r>1 and a monomorphism for r = 1. Furthermore, these are morphisms of functors

Hr+2(G, −) −→ Hr (G, −) and commute with corestrictions. Therefore, using the continuous homomorphism (the augmentation homomorphism) ε :[[FpT ]] −→ Fp,wehaveacommutativedi- agram

ε3 H3(G, [[ FpT ]] ) H3(G)

ψ1([[ FpT ]] ) ψ1(Fp)

H1(G, [[ FpT ]] ) H1(G). ε1

Lemma 10.6.1 If the quotient map μ : T −→ T /G admits a continuous section σ : T /G −→ T , then the map ε1 in the above diagram is a monomorphism.

Proof According to Theorem 10.5.2, ε3 is an isomorphism; and, as we have pointed out, ψ1(Fp) and ψ1([[ FpT ]] ) are monomorphisms. Hence, to prove the claim, it sufﬁces to show that ψ1([[ FpT ]] ) is in fact an isomorphism. Since it is known that it is injective, it is enough to see that it is an epimorphism in addition. To do this observe that (see Proposition 5.5.4 and Theorem 9.1.3) " −1 H3 G, [[ FpT ]] = H3 G, Fpμ (τ) . τ∈T /G

Hence it is enough to see that for each τ ∈ T /G,

−1 −1 −1 ψ1 Fpμ (τ) : H3 G, Fpμ (τ) −→ H1 G, Fpμ (τ) is an epimorphism; we shall show, in fact, that these maps are isomorphisms. 310 10 The Virtual Cohomological Dimension of Proﬁnite Groups

−1 ∼ Fix τ and let T ∈ μ (τ). As we have seen earlier, Hr (NG(T )) = −1 Hr (G, [[ Fpμ (τ)]] ); in fact this isomorphism is the composition

−1 −1 Hn NG(T ), Fp −→ Hn NG(T ), Fpμ (τ) −→ Hn G, Fpμ (τ) , where the ﬁrst map is induced by the NG(T )-homomorphism that embeds Fp naturally in

−1 = \ = ⊗ [[ ]] Fpμ (τ) Fp NG(T ) G Fp [[ FpNG(H )]] FpG , and the second is the corestriction homomorphism (the isomorphism is essentially equivalent to Shapiro’s Lemma; for the dual of the description above, see the proof : −→ of Theorem 6.10.5 in RZ). Consider the restriction fNG(T ) NG(T ) Fp of f to NG(T ). Since, according to the dual of Proposition 10.4.1, the maps ψ−(−) are morphisms of functors and commute with Cor, we deduce that there is a commutative diagram

=∼ −1 H3(NG(T )) H3(G, [[ Fpμ (τ)]] )

−1 ψ1(Fp) ψ1([[ Fpμ (τ)]] )

−1 H1(NG(T )) H1(G, [[ Fpμ (τ)]] ) =∼

Therefore, we need to show that ψ1(Fp) : H3(NG(T )) −→ H1(NG(T )) is an isomorphism. Since G = F T , we deduce that

NG(T ) = GT = NF (T ) × T = CF (T ) × T. (10.16)

Since NF (T ) = CF (T ) is a free pro-p group (cf. RZ, Corollary 7.7.5), we can write = (i) × NG(T ) lim←− F T, i∈I where each F (i) is a free pro-p group of ﬁnite rank. Using again the dual of Propo- sition 10.4.1, we have commutative diagrams (i ∈ I)

(i) H3(NG(T )) H3(F × T)

ψ (F ) (i) 1 p ψ1 (Fp)

(i) H1(NG(T )) H1(F × T) so that = (i) ψ1(Fp) lim←− ψ1 . i∈I 10.6 Pro-p Groups with a Free Subgroup of Index p 311

Recall that Hr (CG(T )) = 0, if r>1, since CG(T ) is free pro-p, and H0(T ) = H2(T ) = Fp = H1(T ) = H3(T ), since T is cyclic of order p (cf. Mac Lane 1995, Theorem IV.7.1). Next we recall the Künneth formula for the homology of a direct product P1 ×P2 of pro-p groups with coefﬁcient module Fp: " Hn(P1 × P2) = Hi(P1) ⊗ Hj (P2). i+j=n

This formula is valid certainly for finite p-groups P1 and P2 (cf. Brown 1982, Corol- lary 5.8). For general pro-p groups, the formula can be verified easily by expressing P1 × P2 as the inverse limit of the finite p-groups P1/U1 × P2/U2, where U1 and U2 run through the open normal subgroups of P1 and P2, respectively. It follows that

(i) (i) (i) H1 F × T = H1 F ⊗ H0(T ) ⊕ H0 F ⊗ H1(T )

∼ (i) (i) (i) = H1 F ⊗ H2(T ) ⊕ H0 F ⊗ H3(T ) = H3 F × T .

(i) × ∞ (i) Note that dimFp H1(F T)< , for the rank of the free pro-p group F is (i) : (i) × −→ (i) × ﬁnite. Since ψ1 (Fp) H3(F T) H1(F T) is injective, according to (i) the dual of Proposition 10.4.1, we infer that ψ1 (Fp) is an isomorphism. Therefore, ψ1(Fp) : H3(NG(T )) −→ H1(NG(T )) is an isomorphism.

Therefore, under the hypotheses of Lemma 10.6.1 (the pro-p group G contains a free pro-p group of index p and μ : T −→ T /G admits a continuous section σ : T /G −→ T )wehavethat " " −1 Cor : H2 NG σ(τ) = H2 G, Fpμ (τ) −→ H2(G) τ∈T /G τ∈T /G is an isomorphism [this uses Theorem 10.5.2 and Proposition 10.5.9(b)], and " " −1 Cor : H1 NG σ(τ) = H1 G, Fpμ (τ) −→ H1(G) τ∈T /G τ∈T /G is an isomorphism [this uses Lemma 10.6.1 and Proposition 10.5.9(b)]. Hence, by Theorem 9.5.1, there exists a free pro-p group P such that G is the free pro-p product

G = NG σ(τ) P. τ∈T /G Thus, using (10.16), we have proved

Proposition 10.6.2 Assume that G is a pro-p group that contains an open subgroup F of index p which is free pro-p. Furthermore, assume that the quotient map 312 10 The Virtual Cohomological Dimension of Proﬁnite Groups

T −→ T /G has a continuous section σ : T /G −→ T (T is the proﬁnite G-space of all subgroups of G of order p). Then G admits a decomposition as a free pro-p product of the form

G = NG σ(τ) P = (Hτ × Tτ ) P, τ∈T /G τ∈T /G where the Hτ arefreepro-p subgroups of F , P is a free pro-p group, and where each Tτ = σ(τ)is a subgroup of G of order p.

Theorem 10.6.3 Assume that G is a second-countable pro-p group that contains an open subgroup F of index p whichisfreepro-p. Let T denote the proﬁnite space of all subgroups of G of order p. Then G admits a decomposition as a free pro-p product of the form

G = (Hτ × Tτ ) P, (10.17) τ∈T /G where {Hτ × Tτ | τ ∈ T /G} is a collection of subgroups of G continuously indexed by T /G such that the Hτ arefreepro-p subgroups of F , each Tτ is a subgroup of G of order p, and P is a free pro-p group. Furthermore, each Hτ is a free factor of F , i.e., F admits a decomposition as a free pro-p product where Hτ is one of the factors.

Proof By Lemma 10.5.1(b) the space T is second-countable; hence the canonical projection T −→ T /G admits a continuous section σ : T /G −→ T (see Sect. 1.3). Thus the decomposition (10.17) follows from Proposition 10.6.2. It remains to show that Hτ is a free factor of F . Since F is a second-countable subgroup of the free pro-p product (10.17) and since it is a normal subgroup of G, we can use Theorem 9.6.2 to deduce that F is a free pro-p product whose g factors are either a free pro-p group P or have the form F ∩ (Hτ × Tτ ) = g g (F ∩ (Hτ × Tτ )) = Hτ , where g ∈ G. Therefore, Hτ is a free pro-p factor of −1 F g = F .

10.7 Counter Kurosh

Let G = G1 ···Gn be a free pro-C product of nontrivial pro-C groups Gi (or more generally a free pro-C product of a sheaf of pro-C groups K), and let H be a closed subgroup of G. A wishful analogue of the Kurosh subgroup structure theorem for abstract groups would say that H can be expressed as a free pro-C product whose factors are either a free pro-C group or intersections of H with conjugates in G of the subgroups Gi . If one assumes that H is an open subgroup, then such a decomposition exists, as we saw in Theorem 7.3.1. But it is easy to see that such a wish is unrealistic for general pseudovarieties of ﬁnite groups C and general closed subgroups of a free pro-C product. 10.7 Counter Kurosh 313

Example 10.7.1 Let p be a prime number and let G1 and G2 be nontrivial pro-p groups. Consider their free profinite product G = G1 G2.Letq be a prime number, q = p, and let Q be a q-Sylow subgroup of G. Note that Q = 1 because G contains a nontrivial free profinite subgroup (for example, the Cartesian subgroup of G:see RZ, Theorem 9.1.6). Observe that Q does not admit a Kurosh-type decomposition: the intersections of Q with conjugates of G1 or G2 are trivial, and Q is not a free profinite group.

Still one may have hoped that if C consists of ﬁnite p-groups, where p is a ﬁxed prime number, such an analogue could be proved. Indeed, as we saw in The- orem 9.6.2, this is the case if one imposes the additional restriction that H is generated by a countable set converging to 1. In this section we construct (in Theo- rem 10.7.4) a counterexample to a possible general analogue of the Kurosh subgroup theorem in the context of free pro-p products. We begin with a technical lemma.

Lemma 10.7.2 Let X and A be proﬁnite spaces, K a proﬁnite group and ω : X × A −→ K a continuous mapping. Put Ax = ω({x}×A) and let M be a closed subset of K. Then ∈ k = −1 ⊆ ∈ x X Ax k Axk M, for some k K is a closed subset of X.

Proof First observe that if S and T are sets and R ⊆ S × T , one has s ∈ S {s}×T ⊆ R = S − πS(S × T − R) , where πS : S × T −→ S is the canonical projection. Next consider the continuous function f : X × A × K −→ K deﬁned by f(x,a,k)= ω(x,a)k (x ∈ X, a ∈ A, k ∈ K). Using continuity and the above observation with S = X × K, T = A and R = f −1(M), we deduce that Y = (x, k) ∈ X × K (x, k) × A ⊆ f −1(M) = ∈ × k ⊆ (x, k) X K Ax M

−1 = X × K − πX×K X × A × K − f (M) ; so Y is a closed subset of X × K (and thus compact), since f −1(M) is closed and πX×K : X × A × K −→ X × K is an open map. Therefore, = ∈ k ⊆ ∈ πX(Y ) x X Ax M, for some k K is compact and so closed.

Recall that a profinite group S acts freely on a pointed profinite space (X, ∗) if it acts continuously on X so that xs = x, for all x ∈ X −{∗}and 1 = s ∈ S, while ∗s =∗, for all s ∈ S (see Sect. 1.3). 314 10 The Virtual Cohomological Dimension of Profinite Groups

Lemma 10.7.3 Let p be a fixed prime number and let C be a cyclic group of order p. Let (X, ∗) be a pointed profinite space on which C acts freely. Let A be K = K another group of order p and consider! the constant! sheaf X(A) over the = K = profinite space X with fiber A. Let K x∈X Ax bethefreepro-p product of this sheaf, and let G = K C = K C be the corresponding semidirect product, as in Proposition 5.7.4. Assume that G can be expressed as a free pro-p product of a family of finite nontrivial p-groups Gy continuously indexed by a profinite space Y :

G = Gy. y∈Y

Then the natural projection ϕ : X −→ X/C admits a continuous section σ : X/C −→ X.

Proof The idea of the proof is to exhibit the existence of a compact subset S of X such that the restriction ϕ|S of ϕ to S is a bijection onto X/C. First, we study the structure of the groups Gy (y ∈ Y). We claim that there exists a y0 ∈ Y such that

ky (i) if y = y0, then Gy = Ax ,forsomeky ∈ K and some xy ∈ X; in particular, in y ∼ this case, Gy ≤ K and Gy = Cp; and = k = k × k =∼ × ∈ (ii) Gy0 A∗,C A∗ C Cp Cp,forsomek K.

To prove this note that, by Corollary 7.1.3, each Ax (x ∈ X) is conjugate in G to a subgroup of some Gy (y ∈ Y).SoK is generated, as a closed normal subgroup of G, by the intersections K ∩ Gy (y ∈ Y). Therefore, by Corollary 5.5.9,

∼ ∼ C = G/K = GyK/K. y∈Y

According to Proposition 5.5.10, there exists y0 ∈ Y such that GyK/K = 1ify = y0 =∼ = ≤ and Gy0 K/K Cp. Therefore if y y0, Gy K, and so (i) is a consequence of Corollary 7.2.3. Now, since G contains a subgroup isomorphic to Cp × Cp, namely = =∼ L A∗,C , we deduce from Corollary 7.1.3 and Lemma 5.7.10(b) that Gy0 Cp × Cp; and so, (ii) follows from Lemma 5.7.10(c). This proves the claim. Let x ∈ X. Then using Corollaries 7.1.3 and 7.1.5 we have that there exists a g −1 unique y ∈ Y such that Ax = g Axg ≤ Gy ,forsomeg ∈ G; so we can deﬁne a function

f : X −→ Y 10.7 Counter Kurosh 315

= ∈ = = c by f(x) y. Note that if c C, then f(xc) f(x) (since Axc Ax ). Hence f induces a function f˜ : X/C −→ Y such that the diagram

f X Y

ϕ f˜ X/C commutes, where ϕ is the natural quotient map. It follows from the claim that f˜ is surjective. We show next that it is also in- g jective. Assume that Ax ≤ Gy so that f(x)= y;sayg = ck (c ∈ C,k ∈ K). Then using the above claim, k G = A y , if y = y ; k = g ≤ ∩ = y xy 0 Axc Ax Gy K ky A∗ , if y = y0. ! = = One then deduces from Corollary 7.1.6 applied to K X Ax that f(x1) f(x2) ˜ if and only if x1 and x2 are in the same orbit under the action of C; i.e., f is injective. Therefore we have proved that f˜ is a bijection. Next we define a section ρ : Y −→ X for the map f as follows: for y ∈ Y define ρ(y) to be the unique x ∈ X such that Gy ∩ K is conjugate to Ax in K. Obviously fρ = idY , and so ρ is a section of f . Our next step is to show that the set S = Im(ρ) is closed in X. To see this first ∩ = ∩ observe that y∈Y (Gy K) ( y∈Y Gy) K is closed in K since y∈Y Gy is closed in G (see Lemma 5.2.1(d)); therefore (see Theorem 5.5.6)

˜ K = Gy ∩ K | y ∈ Y = (Gy ∩ K). y∈Y

Hence, using Corollary 7.1.3, k S = Im(ρ) = x ∈ X ∃k ∈ K,(Gy ∩ K) = Ax for some y ∈ Y = ∈ ∃ ∈ k ≤ ˜ x X k K,Ax K , which is closed by Lemma 10.7.2. Finally, we show that σ = ρf˜ is a continuous section of ϕ. Note that

˜−1 ˜ ˜−1 ˜ ϕσ = f fρf = f IdY f = IdX/C, and so σ is a section. Clearly Im(σ ) = S. Therefore ϕ|S : S −→ X/C is bijective = −1 and continuous. Since S is compact, ϕ|S is a homeomorphism; so σ ϕ|S is continuous. 316 10 The Virtual Cohomological Dimension of Proﬁnite Groups

Theorem 10.7.4 (Kurosh decomposition counterexample) Let p beaﬁxedprime number. Then there exists a closed subgroup G of a free pro-p product of the form (Cp ×Cp)P , with P afreepro-p group, such that G does not admit a decomposition as a free pro-p product of a free pro-p group and conjugates of the intersections of G with Cp × Cp and P .

Proof Throughout this proof the symbol stands for ‘free pro-p product’. Let ∼ C = Cp and let (X, ∗) be a pointed proﬁnite space on which C acts continuously and freely so that the canonical projection X −→ X/C does not admit a continuous section (cf. RZ, Example 5.6.9, where such space is constructed explicitly when p = 2, and Exercise 5.6.10 there; Chatzidakis and Pappas 1992). Consider the free pro-p product

L = A F(X,∗) ∼ of a group A = Cp and the free pro-p group F(X,∗) on the pointed space (X, ∗). Define a homomorphism ρ : C −→ Aut(L) as follows: for c ∈ C, ρ(c) is the continuous automorphism of L that sends A to A identically, and whose restriction to F(X,∗) is the automorphism F(X,∗) −→ F(X,∗) determined by the continuous bijection X −→ X given by x → xc (x ∈ X). Here Aut(L) is assumed to have the compact-open topology. Since C is finite, ρ is continuous. So this defines a continuous action of C on L (cf. RZ, Sect. 5.6). Hence we can construct a corresponding semidirect product

H = L C = A F(X,∗) C, which is a pro-p!group. The closed normal subgroup of L generated by A is the free f pro-p product f ∈F(X,∗) A , according to Proposition 5.7.6 or Corollary 5.3.3. Consider the closed subgroup

K = Ax x∈X of that free product. Observe that c−1Axc = Axc (x ∈ X), and so K is normalized by C. Therefore, K,C=K C = G is the group G considered in Lemma 10.7.3. Thus, because of the way we have chosen the action of C on (X, ∗), G is not a free pro-p product of ﬁnite p-groups. We claim that H = (A × C) P , where P is an appropriate free pro-p group. To show this we proceed in two steps. Let T be the space of subgroups of order p in the group

S = F(X,∗) C. 10.8 Fixed Points of Automorphisms of Free Pro-p Groups 317

By Lemma 5.7.11, every subgroup of S of order p is conjugate to C in S. In other words, the quotient T /S consists of only one point. So the quotient map T −→ T /S obviously admits a continuous section: for example, the map that assigns to the only point of T /S the group C ∈ T . Therefore we may apply Proposition 10.6.2 to obtain a decomposition of S as a free pro-p product

S = NS(C) P, where P is a free pro-p group and NS(C) is the normalizer of C in S. Observe that NS(C) = C by Lemma 5.7.11.SoS = C P . We know that the amalgamated free pro-p product of pro-p groups amalgamating a cyclic group is proper, i.e., it contains the free factors (cf. RZ, Theorem 9.2.4, or more explicitly, Ribes 1971, Theorem 3.1). Using this and the above calculation for S, we check the following isomorphisms

H = A F(X,∗) C = (A × C) C F(X,∗) C

= (A × C) C (C P)= (A × C) P. This proves the claim. Next we show that G cannot be expressed along the lines of the classical Kurosh subgroup theorem; in other words, we show that G cannot be a free pro-p product of a free pro-p group and intersections of G with conjugates of A×C or of P . Assume that this were the case and let M be a nontrivial free factor in such a decomposition. Note that M cannot be a free pro-p group: indeed, if M were free pro-p, it would have quotient finite cyclic groups N of arbitrary large order pn; observe that such N would be a homomorphic image of G (see Corollary 5.1.5); but since G is generated by elements of order p, we have that the order of a finite cyclic quotient of G is at most p. It follows that M must be an intersection of G with a conjugate of A × C; therefore G would be a free pro-p product of finite p-groups. However this is not possible, as explained above.

10.8 Fixed Points of Automorphisms of Free Pro-p Groups

Let L be a proﬁnite group and let ψ : L → L be a continuous endomorphism of L. Put ψ FixL(ψ) = x ∈ L x = x .

Note that FixL(ψ) is a closed subgroup of L.

Theorem 10.8.1 Let p be a prime number and let F be a nonabelian free pro-p group. Let Ψ be a ﬁnite p-group of automorphisms of F and consider the closed subgroup ψ FixF (Ψ ) = FixF (ψ) = x ∈ F x = x, ∀ψ ∈ Ψ ψ∈Ψ 318 10 The Virtual Cohomological Dimension of Proﬁnite Groups of F .

(a) If F is second-countable and Ψ has order p, then FixF (Ψ ) isafreepro-p factor of F , i.e., F admits a decomposition as a free pro-p product of subgroups continuously indexed by a profinite space such that one of those subgroups is FixF (Ψ ). (b) If F has finite rank, then FixF (Ψ ) isafreepro-p factor of F , i.e., there is a free pro-p product decomposition of the form F = FixF (Ψ ) K, for some closed subgroup K of F . As a consequence,FixF (Ψ ) is finitely generated.

Proof Let |Ψ |=pm. Assume that m = 1. Consider the semidirect product G = F Ψ , where Ψ acts naturally on F . Since G is second-countable, one can use Theorem 10.6.3, and hence G admits a free pro-p product decomposition

G = (Hτ × Tτ ) P, τ∈T /G where each Hτ is a free factor of F , each Tτ is a subgroup of G of order p, and P is a free pro-p group. By Corollary 7.1.3 the subgroup Ψ of G is conjugate in = G to one of the groups Tτ ,sayTτ1 . So we may assume that Tτ1 Ψ . Observe that FixF (Ψ ) = CF (Ψ ), the centralizer of Ψ in F . Using Corollary 7.1.6(b), we have = = ∩ = ∩ × = FixF (Ψ ) CF (Ψ ) F CG(Ψ ) F (Hτ1 Ψ) Hτ1 , so that FixF (Ψ ) is a free pro-p factor of F , in this case. This proves part (a). To prove part (b) we proceed by induction on m. Assume that m ≥ 2. Then the center Z of Ψ is nontrivial. Let ψ ∈ Z be an element of order p.Bythe case above FixF ( ψ) is a free pro-p factor of F . Note that Ψ/ ψ is a group of n−1 automorphisms of FixF ( ψ) of order p . Therefore by induction FixF (Ψ ) = FixFixF ( ψ)(Ψ/ ψ ) is a free pro-p factor of FixF ( ψ ). Thus (b) follows.

Corollary 10.8.2 Let p be a prime number and let F be a nonabelian free pro-p group of ﬁnite rank n. Let ψ : F → F be an automorphism of F of order pm, where m is a natural number. Then there exists a free pro-p product decomposition

F = FixF (ψ) K, where K is a closed subgroup of F . In particular,FixF (ψ) has rank at most n.

The assumption in the above corollary that the order of ψ is a power of p is essential in the argument. It allows us to keep our constructions within the category of pro-p groups. The next result shows that if the order of the automorphism is relatively prime to p, the situation is radically different: the subgroup of ﬁxed points must have, in fact, inﬁnite rank.

Theorem 10.8.3 Let p be a prime number and let F beafreepro-p group of rank n ≥ 2. Let α be an automorphism of ﬁnite order m>1 such that m is not a multiple of p. Then the free pro-p subgroup FixF (α) has inﬁnite rank. 10.8 Fixed Points of Automorphisms of Free Pro-p Groups 319

Before proving this result one needs some observations and preliminary lemmas, as well as some notation that we will use in the rest of this section. Let F be a free pro-p group of finite rank n. We view the group of automorphisms (necessarily continuous) Aut(F ) of F as a profinite group (cf. RZ, Corollary 4.4.4). Recall that the Frattini subgroup of F is Φ(F) = F p[F,F], and F/Φ(F) is a vector space of dimension rank(F ) = n over the field Fp with p elements (cf. RZ, Sect. 2.8). If α is an automorphism of F , it naturally induces an automorphism α¯ ∈ Aut(F/Φ(F )) =∼ GL(n, p). Furthermore, the homomorphism

ϕ : Aut(F ) −→ Aut F/Φ(F) given by ϕ(α) =¯α is surjective and its kernel K is a pro-p group of ﬁnite index (cf. RZ, Lemma 4.5.5 and Proposition 7.6.9; Boston 1991). We say that α is a lifting of α¯ .

Lemma 10.8.4 With the above notation, assume that ρ1 ∈ Aut(F/Φ(F )) has order m, where m is not divisible by p. Then ρ1 can be lifted to an automorphism ρ ∈ Aut(F ) of order m.

Proof As pointed out above, there exists some α ∈ Aut(F ) such that ϕ(α) = ρ1. Consider the procyclic subgroup α of Aut(F ) generated by α. Then α is the direct product of its Sylow subgroups (all procyclic); these Sylow subgroups are all ﬁnite, except possibly its p-Sylow group, because K is open and pro-p. Since ∼ α/ α∩K = ρ1, we deduce that there exists an element ρ ∈ α of order m. Clearly ϕ(ρ) = ρ1.

Lemma 10.8.5 Let p be a prime number and let F be a free pro-p group of ﬁnite rank n. Let α and β be automorphisms of F both of ﬁnite order m>1 such that m is not a multiple of p. Assume that α¯ and β¯ are conjugate in Aut(F/Φ(F )). Then α and β are conjugate in Aut(F ).

Proof Since the map ϕ above is an epimorphism, we may assume that α¯ = β¯.This means that β ∈ αK. Consider the closed subgroup of Aut(F/Φ(F ))

K,α= K,β=K α=K β.

By the (proﬁnite) SchurÐZassenhaus theorem (cf. RZ, Theorem 2.3.15), the subgroups α and β are conjugate in the above group. Then αδ = β,forsome δ ∈ K.Soαδ = βr , where r ∈{1,...,m− 1}. Therefore β¯ =¯α = β¯r . It follows that r = 1, i.e., αδ = β.

The next result can be viewed as a weak analogue of Maschke’s Lemma in the context of free pro-p groups.

Lemma 10.8.6 Let p be a prime number and let F be a free pro-p group of finite rank n ≥ 2. Let α be an automorphism of F of finite order m such that p does not 320 10 The Virtual Cohomological Dimension of Profinite Groups divide m. Assume that F = GT (afreepro-p product), where G = FixF (α). Then F = G S, where S is an α-invariant closed subgroup of F , i.e., Sα = S.

Proof Since the kernel of ϕ : Aut(F ) −→ Aut(F/Φ(F )) is a pro-p group, the order of α¯ = ϕ(α) in Aut(F/Φ(F )) is also m. Put V = GΦ(F )/Φ(F ) = G/Φ(G) (cf. RZ, Lemma 9.1.18). By Maschke’s Lemma, the [Fp ¯α]-submodule V of F/Φ(F) has a complement W , i.e., F/Φ(F)= V × W , with W α¯ = W . Since the restriction of α¯ to V is the identity map, the restriction α¯2 of α¯ to W has order m. Then F = GT1, where T1Φ(F)/Φ(F) = W (cf. RZ, Lemma 9.1.18). Replacing T with T1 if necessary, we may assume that F = G T , V = G/Φ(G), W = T/Φ(T), α¯ is the identity map when restricted to V , and the restriction α¯2 of α¯ to W has order m.By Lemma 10.8.4, α¯2 can be lifted to an automorphism α2 of T of order m. Consider the automorphism β of F which is the identity map on G and coincides with α2 on T . Then order(β) = m = order(α) and α¯ = β¯. By Lemma 10.8.5 there exists a −1 γ ∈ Aut(F ) with β = γ −1αγ . Put S = T γ . Then

−1 −1 −1 −1 Sα = T γ αγγ = T βγ = T γ = S,

−1 i.e., S is an α-invariant subgroup of F . Next we show that Gγ = G.Ifx ∈ G = γ −1α βγ−1 γ −1 FixF (α),wehavex = x = x (because β is the identity on G). So −1 −1 Gγ ≤ G. Note that F = Gγ S. Therefore any basis of the free pro-p group −1 −1 Gγ can be extended to a basis of the free pro-p group F ; and since rank(Gγ ) = −1 rank(G), we deduce that dim(GΦ(F )/Φ(F )) = dim(Gγ Φ(F )/Φ(F )), so that −1 −1 GΦ(F ) = Gγ Φ(F). Intersecting with G, we obtain that Gγ Φ(G) = G; there- −1 fore Gγ = G, because Φ(G) consists of the nongenerators of G (cf. RZ, Lemma 2.8.1). Thus F = G S, as desired.

Proof of Theorem 10.8.3 First, we show that it suffices to prove the result when F has finite rank. Indeed, assume the result holds for free groups of finite rank, and suppose that F has infinite rank, while the rank of FixF (α) is finite. Then there exists an open subgroup U of F with U = FixF (α) T (free pro-p product), where T is a closed subgroup of F , necessarily a free pro-p group of infinite rank (cf. RZ, Theorem 9.1.19). Therefore, there exist f1,f2 ∈ F − FixF (α) such that f1,f2 has rank 2. Define = α αm−1 α αm−1 H FixF (α), f1,f1 ,...,f1 ,f2,f2 ,...,f2 . Then H has finite rank, at least 2, and α induces an automorphism on H which is not the identity map and whose order is not divisible by p. However, FixH (α) = FixF (α) has finite rank, a contradiction. From now on we assume that rank(F ) is finite. Put G = FixF (α). By assumption G is a proper subgroup of F . We proceed by induction on m.

Case 1. Assume m = q, a prime different from p. This implies that G = FixF (α) = 1, for otherwise α would act on F element-wise ﬁxed-point-free (i.e., 10.8 Fixed Points of Automorphisms of Free Pro-p Groups 321

F α would be a profinite Frobenius group with kernel F ), and therefore F would be nilpotent (cf. RZ, Corollary 4.6.10), which is absurd because F is a free pro-p group of rank at least 2. Suppose that G is finitely generated. Then there exists an open subgroup U of F containing G and a free pro-p product decomposition U = G T , where T is a closed subgroup of F (cf. RZ, Theorem 9.1.19). We may assume that U is α- m−1 invariant, for otherwise we may replace it with V = U ∩U α ∩···∩U α and apply Theorem 7.3.1 to express the open subgroup V of G T as a free pro-p product, one of whose factors is G. Suppose next that T = 1. Then G is an open subgroup of F ; hence there exists s a natural number s ≥ 1 such that xp ∈ G, for all x ∈ F .LetF˜ = F/[F,F],the abelianized group of F . Note that the induced automorphism α˜ : F˜ → F˜ is not the identity, since otherwise the induced automorphism α¯ on F/Φ(F) would have to be the identity, and this is not possible because, as pointed out above, the kernel of the map Aut(F ) −→ Aut(F/Φ(F )) is a pro-p group, while p does not divide m. s s s Let x ∈ F and denote its image in F˜ by x˜. Since xp ∈ G,wehaveα(˜ x˜p ) =˜xp . s Then (α(˜ x)˜ x˜−1)p = 1; hence α(˜ x)˜ =˜x, because F˜ is a nontrivial free abelian pro- p group. Since x˜ is an arbitrary element of F˜ , this means that α˜ is the identity map, a contradiction. Thus we must have T = 1. By Lemma 10.8.6 we may assume that T is α- invariant. Consider the Cartesian subgroup L of the free pro-p product U = G T , i.e., L is the kernel of the natural epimorphism G T → G × T ; then L is a free pro-p group of infinite rank (cf. RZ, Proposition 8.6.3) and in particular L is not nilpotent. Since L is α-invariant we deduce again from Corollary 4.6.10 in RZ that FixL(α) = 1. But FixL(α) = L ∩ FixF (α) = L ∩ G = 1, a contradiction. Thus G = FixF (α) is infinitely generated when m = q is a prime number.

Case 2. Assume that m = m1q, where q is a prime number and m1 = 1. Define q β = α . Then the order of the automorphism β of F is m1. By the induction hypothesis, FixF (β) is infinitely generated. Observe that FixF (α) ≤ FixF (β). Assume that FixF (α) is finitely generated; then (arguing as at the beginning of the proof) there exist f1,f2 ∈ FixF (β) − FixF (α) so that f1,f2 has rank 2. Hence the rank of the free pro-p subgroup

= α αq−1 α αq−1 H f1,f1 ,...,f1 ,f2,f2 ,...,f2 , FixF (α) is finite and at least 2. Observe that H ≤ FixF (β) and that α restricts to an automorphism of H of order q.ByCase1,FixH (α) is infinitely generated. However, from FixF (α) ≥ FixH (α) ≥ FixF (α) we deduce that FixF (α) = FixH (α), contradicting the assumption that FixF (α) is finitely generated. Thus FixF (α) is infinitely generated, as desired. 322 10 The Virtual Cohomological Dimension of Profinite Groups

10.9 Notes, Comments and Further Reading: Part II

The results in Sects. 7.1 and 7.2 are based on Zalesskii and Melnikov (1989), Za- lesskii (1990), Herfort and Ribes (1985, 1989a, 1989b), Oltikar and Ribes (1979). The first proof of Corollary 7.3.2 (and of Theorem 7.3.1) is due to Binz, Neukirch and Wenzel (1971). For a different approach to this result based on wreath products, see Ribes and Steinberg (2010). For an extension of this result to more general free products, see Gildenhuys and Ribes (1973). In Zalesskii (1992) there is an extension of Theorem 7.3.1 to free pro-C products of infinitely many factors indexed by a profinite space. The structure of closed normal subgroups of free pro-C products is studied in Zalesskii (1995). For necessary conditions on a closed subgroup of a free pro-C product to be prosolvable, see Pop (1995). Proposition 8.1.3 appears in Serre (1980), Proposition I.24, where it is attributed to J. Tits; see also Tits (1970), Proposition 3.2. Some results in Chap. 8 are based on Ribes and Zalesskii (2014) and Ribes and Zalesskii (1996). Part (a) of Proposi- tion 8.4.1 appears in Ribes and Zalesskii (2004). Theorem 9.2.4 appears in Mel’nikov (1989), Theorem 3.7. Assume |T | > 1. Then the hypothesis never holds for n = 0 (just take B to be the trivial G- module Fp). Does it ever hold for n = 1, where Gt = 1fort ∈ T ,ifG is not abelian? If G is the free pro-C product of the Gt , the hypothesis holds for all n ≥ 2 (see The- orem 9.4.3); and in this case the conclusions follow also from Corollaries 7.1.3 and 7.1.5. Are there examples other than free products where the hypothesis holds for some n? More concretely, one may ask

Open Question 10.9.1 Assume that R is a profinite quotient ring of ZCˆ. Let G be a profinite group and let F ={G1,G2} be a set of two closed nontrivial subgroups of G such that for some n ≥ 2 the corestriction map F : ⊕ −→ CorG Hn(G1,B) Hn(G2,B) Hn(G, B) is an isomorphism for every profinite right [[ RG]] -module B. Is it necessarily true that G is the free pro-C product G = G1 G2, for some pseudovariety of finite groups C?

Open Question 10.9.2 Assume that R is a profinite quotient ring of ZCˆ. Let G be a profinite group and let F ={G1,G2} be a set of two closed nontrivial subgroups of G such that −1 (a) whenever r, s ∈{1, 2} and for some g ∈ G, Gs ∩ gGr g contains a nontrivial finite group, then r = s and g ∈ Gr , and −1 (b) whenever H is a finite nontrivial subgroup of G, then H ≤ gGr g , for some g ∈ G and some r ∈ T . Is it necessarily true that for some n ≥ 0 the corestriction map F : ⊕ −→ CorG Hn(G1,B) Hn(G2,B) Hn(G, B) is an isomorphism for every profinite right [[ RG]] -module B. 10.9 Notes, Comments and Further Reading: Part II 323

MayerÐVietoris sequences for profinite groups appear first in Ribes (1969) and Gildenhuys and Ribes (1974), Theorem 1.13. Theorem 9.5.1 appears in the form presented here in Mel’nikov (1989). A slightly less general version of this theorem can be found in Neukirch (1971), where it is stated for cohomology. For a cohomological characterization of free prosolvable products (rather than free pro-p products) see Ribes (1974). Theorem 9.6.1 appears in Ribes (2008). A Kurosh-like description of finitely generated subgroups of free pro-p products was given in Herfort and Ribes (1987). For countably generated subgroups (Theorem 9.6.2) this was done in Haran (1987) and also in Mel’nikov (1989). The result of HerfortÐRibes is very dependent on the finite generation of the subgroup. The introduction of the concept of families of subgroups continuously indexed by a profinite space by Haran, and independently by Mel’nikov, was motivated precisely by a need to describe subgroups (not necessarily finitely generated) of free pro-p products. For a study of p-projective groups acting on C-trees, see Weigel (2009).

M. Hall Pro-p Groups

A group (respectively, a pro-C group) G is said to be an M. Hall group if whenever H is a finitely generated subgroup of G (respectively, a closed finitely generated subgroup of G) there exists a subgroup of finite index (respectively, an open subgroup) U of G containing H such that U = H ∗ L (respectively, U = H L,free pro-C product), where L is some subgroup (respectively, closed subgroup) of U.In Hall (1949) it is proved that free abstract groups are M. Hall groups (see a proof of this and generalizations of it in Sect. 11.2 of this book). In Lubotzky (1982) and Ribes (1991) it is shown that free pro-p groups are also M. Hall groups. In Burns (1971) it is shown that being an M. Hall group is preserved by taking free products of abstract groups; and in Ribes (1991) it is proved that it is also preserved by taking free pro-p products of pro-p groups; see also Shusterman and Zalesskii (2017). Theorem 10.3.1 is due to Serre (1965); the proof in that paper involves some deep algebraic topological tools, such as Steenrod reduced p-th powers. In Serre (1971) the analogous result for discrete groups is proved using purely algebraic methods; and Serre suggests that the same methods would work for pro-p groups. This idea was developed in Haran (1990). Theorem 10.5.2 was proved by Scheiderer (1994). In Scheiderer (1996), Theo- rem 8.1, he proves a result along the same lines without the restriction that G does not contain a subgroup of the form Cp × Cp; his proof involves simplicial cohomology. Theorem 10.6.3 is an analogue of a result of Dyer and Scott (1975) for abstract groups (observe that, unlike the pro-p situation, in the abstract case there is no constraint on the rank of the free group Φ): 324 10 The Virtual Cohomological Dimension of Profinite Groups

Theorem 10.9.3 Let C be a group of prime order p and let Φ be an abstract free group. Assume that R = Φ C. Then R is a free product

R = ∗(Ci × Φi) ∗ L, i∈I where L and each Φi are free groups and the Ci are groups of order p.

Theorem 10.6.3 depends on the second-countability of G, i.e., on G admitting a countable set of generators converging to 1, or more precisely on the existence of a continuous section of the map T → T /G. To see that this in not always guaranteed see RZ, Example 5.6.9. Versions of Theorem 10.6.3 in very special cases appear in Haran (1993), Engler (1995), Herfort, Ribes and Zalesskii (1995, 1998). The idea of using cohomological methods to describe the structure of a pro-p group G with a free pro-p subgroup of index p is due to Scheiderer (1999), where Theorem 10.6.3 is proved for ﬁnitely generated G. This was generalized to the version of this theorem presented here in Herfort, Ribes and Zalesskii (1999) (actually in the latter paper the theorem is stated without assuming the second-countability of G, although the proof does not appear to be valid in that generality).

Open Question 10.9.4 Is the second-countability condition in Theorem 10.6.3 necessary? More explicitly: if G is a pro-p group and it contains an open free pro-p subgroup of index p, can it be expressed as a free pro-p product whose factors are either free pro-p or they have the form C × H , where C is cyclic of order p and H is free pro-p?

Observe that the second-countability condition in the above question is unnecessary if G is torsion-free, because then G is free pro-p according to Serre’s Theo- rem 10.3.1. Theorem 10.7.4 appears in Herfort and Zalesskii (1999). The study of fixed points of automorphisms of a free pro-p group began in Her- fort and Ribes (1990), where Theorem 10.8.3 is proved (when F has finite rank). For an extension to pro-p groups of finite virtual cohomological dimension, see Kochloukova and Zalesskii (2007). For results in the context of abstract groups see Dyer and Scott (1975) and Gersten (1987); for a good survey about fixed points of automorphisms and endomorphisms of free abstract groups, see Ventura (2002). As a complement to Theorem 10.8.3 and Theorem 10.8.1 in this chapter, the following result is proved in Herfort and Ribes (1990).

Theorem 10.9.5 Let p be a prime number, and let m ≥ 1 be an integer dividing p − 1. Consider integers r ≥ 1 and n ≥ 0. Then there exists a free pro-p group F of ﬁnite rank and an automorphism α of F such that FixF (α) has rank n, and the order of α is mp r .

Versions of Theorem 10.8.1 and Corollary 10.8.2 appear in Herfort, Ribes and Zalesskii (1995, 1998, 1999) and Scheiderer (1999). 10.9 Notes, Comments and Further Reading: Part II 325

The following open question appears, in the form of a conjecture, in Herfort, Ribes and Zalesskii (1995). If it holds, it would provide a partial generalization of Corollary 10.8.2.

Open Question 10.9.6 Let p be a prime number and let F beafreepro-p group of finite rank n. Assume that α ∈ Aut(F ) and that the order of the profinite subgroup ∞ α of Aut(F ) is p . Is the rank of FixF (α) finite and at most n? Part III Applications to Abstract Groups

The last five chapters of this book are concerned mainly with abstract groups. Most of the groups that we consider are those that arise as ‘free constructions’ (fundamental groups of graph of groups). The properties that are studied are often properties that hold for the group if and only if they hold for (some of) its finite quotients. For example, one is interested in groups R where conjugacy of two elements a and b occurs if and only if it takes place for their images in the finite quotients of R (conjugacy separability in R). When R is residually finite, this is equivalent to saying that if a and b are conjugate in the profinite completion Rˆ of R, then they are conjugate in R. When R and Rˆ can be interpreted as free constructions (as abstract and profinite groups, respectively), then one can often use the techniques developed in the previous chapters to prove that in fact this property holds for R. Sometimes one can interpret the properties of interest in terms of the profinite (or more generally pro-C) topology of the abstract group R. The second section of Chap. 11 contains a theorem of Marshall Hall that says that if H is a finitely generated subgroup of a free abstract group Φ, then U = H ∗ L, where U is a subgroup of finite index in Φ and L is some subgroup of U.Itis shown that this is in fact equivalent to saying that H is closed in the profinite topology of Φ, i.e., that H is the intersection of the subgroups of finite index in Φ that contain H . A corresponding result holds for other pro-C topologies, when C is an extension-closed pseudovariety of finite groups. One can then deduce that the profinite topology of a finitely generated subgroup H of a free-by-finite abstract group R is precisely the topology induced from the profinite topology of R. In Sect. 11.3 a more general result is proved: if H1,...,Hn is a finite collection of finitely generated closed subgroups of a free abstract group Φ endowed with the pro-C topology, then the product H1 ···Hn is a closed subset of Φ. Section 11.4 records properties of abstract polycyclic-by-finite groups; these groups serve as basic building blocks for the free constructions studied in Chap. 15. Given a prime number p, a free abstract group Φ (with an explicit basis B = {b1,...,bn}) endowed with the pro-p topology, and a finitely generated subgroup H (whose generators are given in terms of B), we describe in Sect. 12.2 an algorithm to find a finite set of generators (written in terms of B) of the closure of H in that topol- 328 ogy. The main result in that section (Theorem 12.2.1) presents a general approach to describing the closure of H inapro-C topology of Φ (when C is extension-closed); one deduces from this theorem what the main difficulties are when trying to make such a description algorithmic. In Sect. 12.3 there are several algorithms of interest in the theory of formal languages on a finite alphabet and in finite monoids; in particular, there is an algorithm that describes how to construct the so-called kernel of a finite monoid, a problem posed by J. Rhodes. For an abstract group R that is either free-by-finite or polycyclic-by-finite, in Chap. 13 we study the relationship between certain constructions in R (normalizers and centralizers of a finitely generated subgroup or intersection of finitely generated subgroups) and corresponding constructions in the profinite completion Rˆ of R.Itis proved, for example, that if H is a finitely generated subgroup of R, the topological ˆ ¯ closure (in R) of the normalizer NR(H ) of H in R coincides with NRˆ (H),the normalizer in Rˆ of the closure of H in Rˆ. In Chap. 14 it is shown that the properties of conjugacy separability and subgroup conjugacy separability are preserved by taking free products of abstract groups. We also show that free-by-finite groups are both conjugacy separable and subgroup conjugacy separable. The basic tools for proving these results are related to the study of minimal invariant subtrees developed in Chap. 8. In Chap. 15 we study how conjugacy separability in groups is preserved under the formation of certain free products with amalgamation. The main result shows that one can construct conjugacy separable groups by forming a free product amalgamating a cyclic subgroup of groups which are either finitely generated free-by-finite or polycyclic-by-finite; in fact one can iterate this process to obtain new conjugacy separable groups. In addition to conjugacy separability we consider in this chapter a whole array of other properties that are preserved by performing amalgamated free products of groups with cyclic amalgamation. The main tools in most results in these chapters are related to the action of abstract groups on abstract trees and the action of profinite groups on profinite trees, and their inter-connections. Chapter 11 Separability Conditions in Free and Polycyclic Groups

11.1 Separability Conditions in Abstract Groups

In this section C is a pseudovariety of ﬁnite groups.

Let R be an abstract group and let N be the collection of all normal subgroups N of R such that R/N ∈ C. As usual we denote by RCˆ the pro-C completion of R, = RCˆ lim←− R/N, N∈N and by ι : R −→ RCˆ the canonical homomorphism of R into RCˆ. The homomorphism ι is injective if and only if N = 1, N∈N i.e., if and only if R is residually C. If this is the case, we think of ι as an inclusion map, so that R ≤ RCˆ; observe that then the topology of RCˆ induces on the subgroup R its pro-C topology; and then we adopt the following notation: if X is a subset ¯ of R, we write X for the topological closure of X in RCˆ, and Cl(X) will denote the topological closure of X in the pro-C topology of R. (There is a certain abuse of language and of notation here; when using this notation it is understood that the pseudovariety C is clearly speciﬁed; but if it is needed for clarity, we shall use the notation ClC(X) for the closure of X in the pro-C topology of R; when writing ‘the’ pro-C topology of a group or ‘its’ pro-C topology—as opposed to ‘a’ pro-C topology on that group—we refer to its full pro-C topology.) Then one has

Lemma 11.1.1 Let R be an abstract residually C group, and let X ⊆ R. Then (a) Cl(X) = R ∩ X¯ , (b) Cl(X) = X¯ , (c) X is closed in the pro-C topology of R if and only if X = R ∩ X¯ .

© Springer International Publishing AG 2017 329 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_11 330 11 Separability Conditions in Free and Polycyclic Groups

Proof Parts (b) and (c) follow from (a). To prove part (a), let y ∈ R ∩ X¯ ;wemust show that y ∈ Cl(X). Every open neighbourhood of y in the pro-C topology of R contains a neighbourhood of the form y(R ∩ U), where U is an open normal subgroup of RCˆ. So it sufﬁces to show that X ∩ y(R ∩ U)= ∅. But this is clear, for X ∩ y(R ∩ U)= X ∩ (R ∩ yU) = X ∩ yU = ∅, since y ∈ X¯ .

Lemma 11.1.2 Let R be an abstract group.

(a) A subgroup H of ﬁnite index in R is open in the pro-C topology of R if and only ∈ C = r if R/HR , where HR r∈R H (the core of H in R). (b) A subgroup H of R is closed in the pro-C topology of R if and only if H = HN. N∈N (c) Assume that R is residually C and a semidirect product R = K H . Then H is closed in the pro-C topology of R. (d) Assume that R is residually C and a free product R = H ∗ T , then H is closed in the pro-C topology of R.

Proof (a) If R/HR ∈ C, then HR is open by definition; hence H is open. Conversely, if H is open, it has finite index in R, and so it has finitely many conjugates. There- fore HR is open and normal; thus R/HR ∈ C. (b) Since the subgroups of the form HN are closed (because they are open), so is their intersection. Conversely, assume that H is a closed subgroup of R, and let x ∈ R − H . Then there exists some open normal subgroup M of R such that xM ∩ H =∅. Hence x/∈ HM; thus H = HN. N∈N (c) Consider the continuous maps ι R ⇒ R, ϕ where ι is the identity map, ϕ(kh) = h(k∈ K,h ∈ H), and R isassumedtohave the pro-C topology. Then H ={r ∈ R | ι(r) = ϕ(r)}. Hence H is closed, since the topology of R is Hausdorff. (d) Let K be the smallest normal subgroup of R containing T . Then R = K H , and so the result follows from part (c).

If H is a subgroup of an abstract group R, the pro-C topology of H is in general ﬁner than the topology on H induced from the pro-C topology of R (cf. Exam- ple 3.1.3 in RZ). We shall often be interested in cases when both topologies on H coincide. The following lemma indicates cases when this happens. 11.1 Separability Conditions in Abstract Groups 331

Lemma 11.1.3 Let R be an abstract group. (a) Assume that R is a semidirect product R = K H . Then the pro-C topology of R induces on H its pro-C topology. (b) Assume that R is a free product R = H ∗ T . Then the pro-C topology of R induces on H its pro-C topology. (c) Assume that C is an extension-closed pseudovariety of ﬁnite groups. Let H be a subgroup of R that is open in the pro-C topology of R. Then the pro-C topology of R induces on H its own pro-C topology.

Proof (a) Let N H with H/N ∈ C (i.e., N is open in the pro-C topology of H ). We need to show that N is open in the induced topology. Now, R/KN =∼ H/N ∈ C, and so KN is open in the pro-C topology of R. Finally, note that KN ∩ H = N. (b) Write R = K H , where K is the smallest normal subgroup of R containing T . Then the result follows from (a). (c) Assume that N H and H/N ∈ C, i.e., that N is open in the pro-C topology of H . It sufﬁces to show that N is open in the pro-C topology of R.By Lemma 11.1.2(a) this is equivalent to showing that R/NR ∈ C. To prove this, consider the subgroup K = HR ∩ N of R. Then

H/K ≤ H/HR × H/N ∈ C, ∈ = t ri since H is open in R. Choose r1,...,rt R so that KR i=1K . Observe that ∼ HR/K = HRN/N ≤ H/N ∈ C, r and so HR/K i ∈ C, for all i.Now,

r1 rt HR/KR ≤ HR/K ×···×HR/K ∈ C.

Thus the extension R/KR of HR/KR by R/HR belongs to C. Finally, note that NR = KR and thus R/NR ∈ C, as needed.

Recall that when C is the class of all finite groups, we write residually finite, ˆ instead of residually C; and, as usual, we write R, rather than RCˆ, for the profinite completion of R.LetR be an abstract group, and let N ={N | N f R} denote the collection of its normal subgroups of finite index. R is termed subgroup separable or LERF if every finitely generated subgroup H of R is closed in the profinite topology of R, i.e., if H = HN = {M | H ≤ M ≤f R}. N∈N Clearly subgroup separable groups are residually finite. So, with the notation established above, saying that R is subgroup separable means that for every finitely generated subgroup H of R one has H = Cl(H ).

Lemma 11.1.4 Let R be a subgroup separable abstract group. (a) Let H be a finitely generated subgroup of R. Then the profinite topology of H coincides with the topology induced by the profinite topology of R. 332 11 Separability Conditions in Free and Polycyclic Groups

(b) If H and K are ﬁnitely generated subgroups of R and HN/N = KN/N for all N f R, then H = K.

Proof (a) We need to show that a subgroup N of H of finite index is open in the topology induced by the profinite topology of R. Since H is finitely generated, so is N. Hence N is closed in the profinite topology of R, and therefore in the topology induced on H . But since N has finite index in H , this means that it is also open in that topology. ¯ ¯ ˆ (b) Since HN/N = KN/N for all N f R, we deduce that H = K in R (see Sect. 1.2). Therefore, using Lemma 11.1.1(a), H = Cl(H ) = R ∩ H¯ = R ∩ K¯ = Cl(K) = K.

We turn now to a different kind of separability condition. Let C be a pseudovariety of finite groups. An abstract group R is called conjugacy C-separable if for any pair of elements x,y ∈ R, these elements are conjugate in R if and only if their images in every finite quotient of R which is in C are conjugate, or equivalently, if x = yr for every r ∈ R, then there exists some N R with R/N ∈ C such that xN = xsN for every s ∈ R.IfC is the class of all finite groups, we simply write conjugacy separable, rather than conjugacy C-separable. If R is conjugacy C- separable, then it is residually C: indeed, if 1 = x ∈ R and x ∈ N for every N R with R/N ∈ C, then x and 1 are conjugate mod N for every N R with R/N ∈ C, while obviously x and 1 are not conjugate in R.

Lemma 11.1.5 Let R be a residually C abstract group. Then the following conditions are equivalent: (a) R is conjugacy C-separable. (b) For any pair of elements x,y ∈ R, if x and y are conjugate in RCˆ, then they are γ conjugate in R: if x = y , for some γ ∈ RCˆ, then there exists some r ∈ R with x = yr . (c) For every x ∈ R, its conjugacy class xR = xr = r−1xr r ∈ R in R is closed in the pro-C topology of R. (d) For every x ∈ R, xR = R ∩ xRCˆ .

Proof Let N be the collection of all normal subgroups N of R such that R/N ∈ C. Let x,y ∈ R and assume that xN and yN are conjugate in R/N for every N ∈ N . t Let TN be the set of all elements t of R/N such that xN = (yN) ; then TN = ∅, and if N1,N2 ∈ N with N1 ≤ N2, the canonical homomorphism R/N1 −→ R/N2 { | ∈ N } sends TN1 into TN2 . Hence TN N is an inverse system of ﬁnite nonempty sets. Therefore (see Sect. 1.1) its inverse limit is nonempty. Let ∈ ⊆ = γ lim←− TN lim←− R/N RCˆ. N∈N N∈N 11.2 Subgroup Separability in Free-by-Finite Groups 333

γ Hence x = y in RCˆ. This shows that x and y are conjugate in RCˆ if and only if they are conjugate mod N for every N ∈ N . Thus, (a) ⇐⇒ (b). Conditions (b) and (d) are obviously equivalent. Finally, observe that xR = xRCˆ R RCˆ ¯ since x and x have the same images in R/N = RCˆ/N, for every N ∈ N (see Sect. 1.2). Hence the equivalence of (c) and (d) follows from Lemma 11.1.1(c).

An abstract group R is subgroup conjugacy C-separable if whenever H and K are finitely generated subgroups of R and closed in its pro-C topology, then H and K are conjugate in R if and only if their images in every quotient group R/N ∈ C of R are conjugate. Observe that such groups must be residually C; indeed, let N be the collection of all normal subgroups N of R such that R/N ∈ C; define S = N N∈N to be the C-residual of R;forx ∈ S,ClC( x) ≤ S, because S is closed in the pro-C topology of R; since the subgroups ClC( x) and 1 are conjugate modulo N,for every N ∈ N , we deduce that ClC( x) is conjugate to 1, and so x = 1. For groups that are residually C, one has the following characterization of subgroup conjugacy C-separability: whenever H and K are finitely generated subgroups of R that are closed in its pro-C topology, then H and K are conjugate in R if and only if H¯ ¯ and K are conjugate in RCˆ.IfC is the pseudovariety of all finite groups, we simply write subgroup conjugacy separable. Note that in the latter definition we require that H and K be closed in the profinite topology of R (in addition to being finitely generated); of course this requirement is automatically satisfied if R is subgroup separable. Finally, a subgroup H of an abstract residually C group R is said to be conjugacy C-distinguished if whenever y ∈ R, then y has a conjugate in H if and only if the same holds for the images of y and H in every quotient group R/N ∈ C of R,or equivalently, yR ∩ H =∅if and only if yRCˆ ∩ H¯ =∅.IfC is the pseudovariety of all finite groups, we simply write conjugacy distinguished.

11.2 Subgroup Separability in Free-by-Finite Groups

Let C be an extension-closed pseudovariety of finite groups and let R be an abstract group. Recall that R is a free-by-C group if it contains a normal abstract free subgroup Φ with R/Φ ∈ C. When C is the pseudovariety of all finite groups, we write instead free-by-finite group or virtually free group. In this section we begin the study of the pro-C topology of free-by-C groups. We start with the following observation.

Lemma 11.2.1 Let C be an extension-closed pseudovariety of ﬁnite groups and let R be a free-by-C group. Then R is residually C.

Proof Say Φ is a normal free subgroup of R with R/Φ ∈ C. It sufﬁces to prove that if 1 = x ∈ Φ, then there exists an N R with R/N ∈ C and x/∈ N. Since C is 334 11 Separability Conditions in Free and Polycyclic Groups extension-closed, Φ is residually C (cf. RZ, Proposition 3.3.15). Let M Φ be such ∈ C ∈ = r ∈ that Φ/M and x/M. Choose N r∈R M , the core of M in R; then x/N. Since C is extension-closed and (R/N)/(Φ/N) = R/Φ,wehaveR/N ∈ C.

Theorem 11.2.2 Free-by-ﬁnite abstract groups are subgroup separable.

Proof Let R be an abstract group that contains a free normal subgroup Φ of finite index. We claim that we may assume that R = Φ is a free group. Indeed, note that the profinite topology on R induces on Φ its own profinite topology since Φ has finite index in R (see Lemma 11.1.3(c)). Let H be a finitely generated subgroup of R; then H ∩ Φ is also finitely generated. If we assume known that H ∩ Φ is closed in the profinite topology of Φ, it follows that it is closed in the profinite topology of R, since Φ is closed in R. Hence each coset g(H ∩ Φ) is closed in R. Since H is the union of finitely many cosets of H ∩ Φ,itisclosedinR. This proves the claim. Therefore we shall prove the result for R = Φ = Φ(X), a free abstract group with abs basis X. Interpret Φ as the fundamental group π1 (, v) of the graph consisting of a single vertex v and the set of different edges (loops) {ex | x ∈ X} indexed by X (see Example A.2.7 in Appendix A). Let {h1,...,hr } be a set of nontrivial generators of H and let a ∈ Φ − H be an element of Φ which is not in H . It suffices to prove that there exists a subgroup U of Φ of finite index such that a/∈ U and U ≥ H . ∈{ } = ε1 ··· εs ∈{ | ∈ } =± Let h h1,...,hr . Say h e1 es (ei ex x X ,εi 1). Construct ¯ ˜ a corresponding graph Γh in the form of an ‘s-polygon’ as follows: Let ν : −→ be the abstract universal covering graph of (see Appendix A, Sect. A.2). Fix a vertex v˜ of ˜ and let h˜ be the unique path starting at v˜ which lifts h (see Proposi- ˜ ˜ tion A.2.2(a)). Let Γh be the minimal subtree of containing all the vertices of h; this is a graph with exactly two ‘end-vertices’ (vertices of valency 1) v˜ and w˜ , and ¯ whose other vertices have valency 2: a ‘line’. Let Γh be the graph obtained from Γh ¯ ¯ by identifying v˜ and w˜ ; denote by v¯h this vertex in Γh.Letν¯h : Γh −→ be the morphism of graphs induced by the restriction νh : Γh −→ of ν to Γh. For a ∈ Φ − H , we construct a finite subtree Γa and a morphism νa : Γa −→ similarly. Define a finite graph Γ to be the result of identifying the vertices v,˜ v¯1,...,v¯r to a vertex v¯ in the disjoint union of the ‘line’ Γa and the polygons ¯ ¯ Γh1 ,...,Γhr . [Rough description: Γ is the union of a bouquet of s finite polygons ¯ ¯ : −→ Γhi and a finite ‘line’ Γa joined at a vertex v.] Denote by γ Γ the mor- { } ¯ = phism of graphs determined by νa,νh1 ,...,νhs . Note that γ(v) v. It follows abs from this construction that γπ (Γ, v)¯ = H and γ(Γa) = a. Since Γ is finite, there 1 is a factorization γ = γ fr ···f1, where γ : Γ −→ is an immersion and each fi is a folding (see Proposition A.3.1(b)). Put v¯ = fr ···f1(v)¯ . By Proposition A.3.2, γ π abs(Γ, v¯ ) = H ; furthermore, since a/∈ H , the images in Γ of the end-points of 1 Γa do not coincide, i.e., the image Γa of Γa in Γ does not determine a circuit. By Lemma A.2.9, there exists a covering γ¯ : Γ¯ −→ such that Γ is a sub- ¯ = ¯ ¯ =¯ abs ¯ graph of Γ , V(Γ ) V(Γ) and γ extends γ . Put U γπ1 (Γ,v). Clearly 11.2 Subgroup Separability in Free-by-Finite Groups 335

≤ ∈ H U. Moreover, if a U, we would have that Γa determines a circuit by Proposi- ∈ = abs |¯−1 | tion A.2.2(a). Hence a/U. Finally, the index of U in Φ π1 (, v) is γ (v) , according to Proposition A.2.6, and so it is ﬁnite since Γ¯ has ﬁnitely many vertices.

In fact the proof of the above theorem shows something (apparently) stronger; namely one has the following result.

Theorem 11.2.3 If H is a ﬁnitely generated subgroup of a free abstract group Φ, then there is a subgroup U of ﬁnite index in Φ that contains H as a free factor.

Indeed, continuing with the notation in the above proof, by Corollary A.1.5 abs abs ¯ ¯ π1 (Γ ,v ) is a free factor of π1 (Γ,v), and by Proposition A.2.3 γ and γ are injective on fundamental groups. One can also see the above theorem by appealing (in the presence of Theo- rem 11.2.2) to the following more general result. Recall that if C is a pseudovariety of finite groups, then an abstract group R is residually C if its pro-C topology is Hausdorff. If R = Φ is a free abstract group, then Φ is residually C if, for example, C is a pseudovariety of finite groups which contains all finite p-groups, for a certain fixed prime number p; in particular, if C is the pseudovariety of all finite groups, or any extension-closed pseudovariety of finite groups (cf. RZ, Proposition 3.3.15).

Theorem 11.2.4 Let Φ be an abstract free group. Let C be a pseudovariety of finite groups such that Φ is residually C. Endow Φ with the pro-C topology. (a) Let K be a closed subgroup of Φ and assume that H is a finitely generated subgroup of Φ which is a free factor of K. Then there exists an open subgroup U of Φ containing K such that H is a free factor of U. (b) Let H be a finitely generated closed subgroup of Φ. Then there exists an open subgroup U of Φ containing H such that U = H ∗ R for some subgroup R of U.

Proof Part (b) is a special case of (a): just put K = H . We prove (a). Say Φ = Φ(X) has a basis X.LetΓ = Γ(Φ,X)be the Cayley graph of Φ with respect to X. = abs \ We know that H π1 (H Γ)(see Corollary A.2.8). Since H is ﬁnitely generated, there exist a ﬁnite number of cycles in H\Γ based at H 1 representing the generators of H . Denote by the subgraph of H\Γ underlying the union of those cycles. Then abs = abs \ = π1 () π1 (H Γ) H.

Let 1 be the image of in K\Γ under the mapping H \Γ −→ K\Γ . Then from abs \ −→ abs \ the injectivity of π1 (H Γ) π1 (K Γ)we deduce that H is naturally a sub- abs group of π1 (1). Since by assumption H is a free factor of K, it follows from the abs Kurosh subgroup theorem for abstract groups that H is a free factor of π1 (1). 336 11 Separability Conditions in Free and Polycyclic Groups

By Lemma 8.1.1 there exists an open subgroup U of Φ containing K such that 1 maps injectively into U\Γ . So we may consider 1 as a subgraph of U\Γ . There- abs abs \ = fore π1 (1) is a free factor of π1 (U Γ) U (see Corollary A.1.5). Thus H is a free factor of U.

Corollary 11.2.5 Let C be an extension-closed pseudovariety of ﬁnite groups. Let R be an abstract free-by-C group. Let H be a ﬁnitely generated closed subgroup of R. Then the topology induced on H by the pro-C topology of R is the pro-C topology of H .

Proof Say Φ is an open subgroup in the pro-C topology of R which is an abstract free group. By Theorem 11.2.4 above, H ∩ Φ is a free factor of an open subgroup U of Φ;sayU = (H ∩ Φ)∗ L. By Lemma 11.1.3(c), the topology induced on U by the pro-C topology of R is the pro-C topology of U. By Lemma 11.1.3(b), the pro-C topology of U induces the pro-C topology of H ∩ Φ. Since H ∩ Φ is open in H ,the result follows.

Corollary 11.2.6 Let R be a free-by-finite abstract group. Then the profinite topology of a finitely generated subgroup H of R coincides with the topology induced on H by the profinite topology of R.

Proof This follows from Theorem 11.2.2 and Corollary 11.2.5.

Example 11.2.7 (Inﬁnitely generated closed subgroups) Let Φ = Φ(X) be an abstract free group. If N is a normal subgroup of Φ and C is a pseudovariety of ﬁnite groups, then saying that N is closed in the pro-C topology of Φ amounts to saying that R/N is residually C. For example, one has

(a) Any subgroup of the form γi(Φ) (the i-th term of the lower central series) is closed in the pro-p topology (and so in the proﬁnite topology) of Φ, for every prime number p (cf. Gruenberg 1957, Theorem 2.2).

(b) (A generalization of (a)) Let W be a collection of words w(xi1 ,...,xin(w) ) in the elements of the basis X. Recall that the verbal subgroup W(Φ) of the group Φ corresponding to W is the subgroup generated by the elements of Φ of the form w(g1,...,gn(w))(w∈ W,g1,...,gn(w) ∈ Φ) (cf. Kargapolov and Merzljakov 1979, Sect. 15). Then one has:

If Φ(X)/W Φ(X) is nilpotent, then it is residually finite. (Φ(X)/W(Φ(X)) is the free group in the pseudovariety determined by W .) In- deed, let 1 = f W(Φ(X)) ∈ Φ(X)/W(Φ(X)), with f ∈ Φ(X), and let Y be a finite subset of X such that f ∈ Φ(Y) ≤ Φ(X). Consider the homomorphism ϕ : Φ(X) −→ Φ(Y) determined by x → x,ifx ∈ Y , and x → 1, if x ∈ X − Y . Since ϕ−1(W(Φ(Y ))) = W(Φ(X)), we deduce that ϕ(f)W(Φ(Y)) = 1in Φ(Y )/W(Φ(Y )). Since Φ(Y )/W(Φ(Y )) is finitely generated nilpotent, it is residually finite (see Theorem 11.4.1), and so there is a homomorphism into a finite group that sends ϕ(f)W(Φ(Y)) to a nontrivial element; therefore the same is true for f W(Φ(X)) in Φ(X)/W(Φ(X)). 11.3 Products of Subgroups in Free Abstract Groups 337

(c) Any subgroup of the form Φ(i) (the i-th term of the derived series) is closed in the pro-p topology (and so in the profinite topology) of Φ, for every prime number p (cf. Gruenberg 1957, Theorem 8.3). (d) Let R be an abstract group that contains a free abstract group Φ = Φ(X) as a normal subgroup of finite index. Then it follows from (a), (b) or (c) that the (i) subgroups of the form γi(Φ), Φ or W(Φ(X)), where W(Φ(X)) satisfies the assumptions of (b), are closed in the profinite topology of R.

11.3 Products of Subgroups in Free Abstract Groups

Throughout this section we assume that C is an extension-closed pseudovariety of ﬁnite groups.

We have seen in Theorem 11.2.2 that a finitely generated subgroup of a free abstract group Φ is closed in the profinite topology of Φ. In Theorem 11.3.6 we extend this result to show that the product of finitely many finitely generated subgroups of Φ is also a closed subset. In fact, we prove this in a more general context: this is true if we replace the free group Φ with a virtually free group and the profinite topology with a pro-C topology.

Proposition 11.3.1 Let Φ be a free abstract group and let H be a ﬁnitely generated subgroup of Φ. Then

rank Cl(H ) ≤ rank(H ) (recall that Cl(H ) is the closure of H in the pro-C topology of Φ).

Proof Recall that throughout this book we use the notation rank(T ) only when T is a free group (abstract or pro-C); and for a proﬁnite group T we use the notation d(T) to mean the smallest cardinal of a set of topological generators of T converging to 1. Suppose that rank(Cl(H )) > rank(H ), and let rank(H ) = n.Letx1,x2,...,xn+1 be a subset of a basis of the free group Cl(H ), and denote by K the subgroup generated by x1,x2,...,xn+1. Then Cl(H ) = K ∗ R, for some subgroup R of Cl(H ). By Theorem 11.2.4(a), there exists some open subgroup U of Φ containing Cl(H ) such that U = K ∗ L, and so

UCˆ = KCˆ LCˆ, where denotes the free pro-C product, and UCˆ, KCˆ and LCˆ the pro-C completions of U, K and L, respectively. Moreover, since U is open in the pro-C topology of Φ, the topology on U induced from the pro-C topology of Φ coincides with its own pro-C topology (see Lemma 11.1.3(c); note that it is here where we need that C be ¯ extension-closed). It follows that UCˆ = U. Since K is a free factor of U, we deduce ¯ ¯ ¯ that K = KCˆ (see Lemma 11.1.3(b)). Then one has that KCˆ = K ≤ Cl(H ) = H . Hence the composition of natural maps ¯ ¯ Cl(H ) = H→ U −→ KCˆ 338 11 Separability Conditions in Free and Polycyclic Groups

¯ ¯ is an epimorphism. Thus, d(H) ≥ rank(KCˆ) = n + 1. However, d(H) ≤ rank(H ) = n, a contradiction.

Remark 11.3.2 The inequality rank(Cl(H )) ≤ rank(H ) can be strict. For example, if H is a nonopen (in the pro-C topology) subgroup of ﬁnite index in a free abstract group Φ of ﬁnite rank.

Corollary 11.3.3 With the above notation, if rank(H ) ≤ 2, then rank(Cl(H )) = rank(H ).

Proof For rank(H ) = 1, this is clear, and if rank(H ) = 2, just observe that Cl(H ) cannot be abelian.

Corollary 11.3.4 Let R be an abstract free-by-C group and let H be a ﬁnitely generated subgroup of R. Then the minimal number of generators d(Cl(H )) of the closure Cl(H ) of H in the pro-C topology of R is ﬁnite.

Proof LetΦ be a normal free subgroup of R such that R/Φ ∈ C. Note that Cl(H ) = xCl(H ∩ Φ), where x ranges over a set of representatives of the cosets of H ∩ Φ in H (observe that according to Lemma 11.1.3(c), the notation Cl(H ∩ Φ) in unambiguous: it does not matter whether one takes the closure in R or in Φ). Hence the minimal number of generators d(Cl(H )) of Cl(H ) is bounded by rank(H ∩ Φ)+[R : Φ], by Proposition 11.3.1.

Let R be an abstract group endowed with the pro-C topology and let n ≥ 2 be a natural number. We say that R is n-product subgroup separable if whenever H1,...,Hn are ﬁnitely generated closed subgroups of R, then H1 ···Hn is a closed subset of R. This is of course a property of R that depends on C, which it is assumed to be ﬁxed.

Lemma 11.3.5 Let R be an abstract group endowed with its pro-C topology. (a) Let S be a closed subgroup of R whose pro-C topology coincides with the topology induced by the pro-C topology of R. If R is n-product subgroup separable, then so is S. (b) If R is n-product subgroup separable, then so is any open subgroup of R. (c) If R contains an open subgroup which is n-product subgroup separable, then so is R.

Proof Part (a) is obvious. Part (b) follows from (a) and Lemma 11.1.3. To prove (c) assume that U is an open subgroup of R which is n-product subgroup separable. = r Then the core UR r∈R U of U in R is n-product subgroup separable as well by (b). Hence, replacing U by UR, if necessary, we may assume that U is open and normal in R.LetH1,...,Hn be finitely generated closed subgroups of R.We prove, by induction on the number of Hi that are not contained in U, that H1 ···Hn 11.3 Products of Subgroups in Free Abstract Groups 339 is closed in the pro-C topology of R.IfHi ≤ U for all i = 1,...,n, the result is ∩ clear. Since each Hi is finitely generated and U Hi has finite index in Hi ,wehave ∩ ≤ = . ∩ that U Hi is also finitely generated. Pick Ht U. Write Ht j hj (U Ht ), (hj ∈ Ht ). Therefore ··· = . hj ··· hj ∩ ··· H1 Hn hj H1 Ht−1(U Ht )Ht+1 Hn (a finite union). j

hj ··· hj ∩ ··· By the induction hypothesis, H1 Hi−1(U Hi)Hi+1 Hn is closed in R. Thus H1 ···Hn is closed in R.

Theorem 11.3.6 Let R be an abstract free-by-C group. Let K,H1,...,Hm (m ≥ 0) be ﬁnitely generated subgroups of R which are closed in its pro-C topology; then the set

H1 ···HmK is closed in the pro-C topology of R. In other words, R is n-product subgroup separable, for all n.

Proof By Lemma 11.3.5 we may assume that R = Φ is an abstract free group. Since the subgroups K,H1,...,Hm are finitely generated, there is a free factor of Φ of finite rank that contains all of them. Since the pro-C topology of Φ induces the full pro-C topology on a free factor of Φ (see Lemma 11.1.3(b)), we may assume that Φ has finite rank. By Theorem 11.2.4 there is an open subgroup U of Φ such that U = K ∗ L, for some subgroup L of U. So, using Lemma 11.3.5 again, we may assume that K is a free factor of Φ. Hence, taking pro-C completions, ¯ ¯ ΦCˆ = K L, ¯ ¯ since K = KCˆ and L = LCˆ (see Lemma 11.1.3(b)). Let Y be a basis of K as an abstract free group and let X be a basis of Φ contain- ¯ ing Y . Note that X and Y are also bases of the free pro-C groups ΦCˆ and K = KCˆ, respectively. Let Γ(Φ) and Γ(K) denote the abstract Cayley graphs of Φ and K ¯ with respect to X and Y , respectively; and let Γ(ΦCˆ) and Γ(K) denote the profinite ¯ Cayley graphs of ΦCˆ and K with respect to X and Y , respectively. Then we have a commutative diagram of graphs

Γ(Φ) Γ(ΦCˆ)

Γ(K) Γ(K)¯ ¯ ¯ ¯ Note that H1 ···HmK = H1 ···HmK, since the group ΦCˆ is compact. So, by Lemma 11.1.1(c), proving that H1 ···HmK is closed in the pro-C topology of Φ is equivalent to proving that ¯ ¯ ¯ H1 ···HmK ∩ Φ = H1 ···HmK. 340 11 Separability Conditions in Free and Polycyclic Groups

¯ ¯ Hence, if hi ∈ Hi , k ∈ K and

h1 ···hmk ∈ Φ, we need to prove that h1 ···hmk ∈ H1 ···HmK. We do this by induction on m.Ifm = 0, this means that K¯ ∩ Φ = K, which is just the assumption that K is closed in the pro-C topology of Φ. Assume that m ≥ 1 and that the result holds when the number of factors of the form hi is less than m. Since h1 ···hmk ∈ Φ, the chain [1,h1 ···hmk] in Γ(ΦCˆ) [ −1 ··· −1 ] ∈ ¯ is finite, and hence so is hm h1 ,k . Since k Γ(K), there exists a vertex v [ −1 ··· −1 ] ∈ ¯ [ −1 ··· −1 ] of hm h1 ,k such that v Γ(K) and hm h1 ,v has minimal cardinality −1 ··· −1 (alternatively, if we consider the path from hm h1 to k whose underlying graph [ −1 ··· −1 ] ¯ = is hm h1 ,k , then v is the first vertex of that path which is in Γ(K)). If v −1 ··· −1 −1 ··· −1 ∈ ¯ ··· ∈ ¯ ∩ = hm h1 , then hm h1 K. Therefore h1 hmk K Φ K, and we are done. [ −1 ··· −1 ] So, assume that the finite chain hm h1 ,v has at least one edge. Observe that, by the definition of the Cayley graph Γ(K)¯ , v = kr for some r ∈ K (since [v,k] is finite). Hence h1 ···hmv ∈ H1 ···HmK if and only if h1 ···hmk ∈ H1 ···HmK. Therefore, from now on we may assume that k = v. ∈[ −1 ··· −1 ] Next note that v hm h1 , 1 . Indeed, let e be the edge of the chain [ −1 ··· −1 ] ∈ ¯ ∈[ −1 ··· −1 ] hm h1 ,v such that e/Γ(K) but v is a vertex of e. Then e hm h1 , 1 , for otherwise collapsing the C-subtree

−1 ··· −1 ∪[ ] hm h1 , 1 1,v of the C-tree

−1 ··· −1 ∪[ ]∪ −1 ··· −1 hm h1 , 1 1,v hm h1 ,v [ −1 ··· −1 ] to a point, we get a circuit corresponding to hm h1 ,v since e is not in the subtree (for it is not in Γ(K)¯ ). But this contradicts the fact that collapsing a C- subtree of a C-tree produces a tree (see Lemma 2.4.7). Hence

= ∈ −1 ··· −1 v k hm h1 , 1 . For each i = 1,...mdeﬁne ¯ Li = Hi[1,rj ], j where the collection of the rj is a ﬁnite set of generators of Hi . Then Li is a C- subtree of Γ(ΦCˆ) (see the reasoning in the proof of Proposition 8.2.4). We deduce from Proposition 2.4.3 that = −1 ··· −1 ∪ −1 ··· −1 ∪···∪ T hm h2 L1 hm h3 L2 Lm C ∈[ −1 ··· −1 ]⊆ is a -subtree of Γ(ΦCˆ). Moreover, k hm h1 , 1 T . 11.3 Products of Subgroups in Free Abstract Groups 341

∈ −1 ··· −1 = −1 ··· −1 = Case 1. k hm hi+1Li ,forsomei>1 (convention: if i m, hm hi+1 1). Then we shall prove the result by induction. As pointed out above, it is true if the total number of factors is m + 1 = 1; so assume that m + 1 > 1 and that the result is true if the total number of factors is less than m + 1. Now, we have

··· = ∈ ¯ ∈ hi+1 hmk hif some hi Hi,f Φ . So −1 ··· ∈ ∩ ¯ ··· ¯ ¯ hi hi+1 hmk Φ Hi HmK. Since i>1, the number of factors in this product is less that m + 1, and so by induction −1 ··· ∈ ··· hi hi+1 hmk HiHi+1 HmK. Next, since ··· = ··· −1 ··· ∈ h1 hmk h1 hihihi hi+1 hmk Φ, we have that ··· ∈ h1 hihi Φ. ··· ∈ ··· And again by the induction hypothesis, we have h1 hihi H1 Hi . Thus,

h1 ···hmk ∈ H1 ···HmK. ∈ −1 ··· −1 Case 2. k hm h2 L1. [ ]∩ −1 ··· −1 ∩ −1 ··· −1 ∪···∪ = ∅ We claim that k,1 hm h2 L1 (hm h3 L2 Lm) . To see this note ﬁrst that since [k,1]⊆T ,wehave

[ ]= [ ]∩ −1 ··· −1 ∪ [ ]∩ −1 ··· −1 ∪···∪ k,1 k,1 hm h2 L1 k,1 hm h3 L2 Lm . If the claim were not correct, we would have that [k,1] is the disjoint union of two closed subgraphs, and this would contradict the connectedness of [k,1] (see Lemma 2.1.9). This proves the claim. Consider a vertex

∈[ ]∩ −1 ··· −1 ∩ −1 ··· −1 ∪···∪ k k,1 hm h2 L1 hm h3 L2 Lm . ≥ ∈ −1 ··· −1 Then for some i 2, we have k hm hi+1Li . Put ¯ = −1 ··· −1 ¯ ··· H hm h2 H1h2 hm. ¯ ˜ = −1 ··· −1 ¯ ∩ ¯ C = ˜ ∩ ¯ Then H acts on L1 hm h2 L1.SoH K acts on the -tree S L1 Γ(K). ¯ ˜ Note that the quotient graph H\L1 is ﬁnite. Consider the quotient map of graphs ˜ ¯ ˜ ψ : L1 −→ H\L1. It induces an epimorphism of graphs ψ˜ : H¯ ∩ K¯ \S −→ ψ(S). 342 11 Separability Conditions in Free and Polycyclic Groups

Observe that ψ˜ is an isomorphism: indeed, if x ∈ H¯ and s and s are either both in V(S)or both in E(S) with xs = s , then x ∈ K¯ . Hence H¯ ∩ K¯ \S is ﬁnite. Thus the morphism of graphs S −→ H¯ ∩ K¯ \S has a 0-transversal J with k as one of its a vertices (see Proposition 3.4.5). Since k ∈ S, there exists a g ∈ H¯ ∩ K¯ such that gk ∈ J. Since J is ﬁnite and S ⊆ Γ(K)¯ , there exists an r ∈ K with k r = gk. Then −1 −1 h1 ···hmk = h1 ···hmg gk = h1 ···hmg k r. −1 ¯ Therefore, h1 ···hmg k ∈ Φ; and since g ∈ H ,say = −1 ··· −1 ··· g hm h2 h1h2 hm, ∈ ¯ for some h1 H1,wehave

··· −1 = ··· −1 ··· −1 ··· −1 = ··· ∈ h1 hmg k h1 hm hm h2 h1h2 hm k h1h2 hmk Φ, ∈ ¯ ∈ −1 ··· −1 where h1 H1. So, since k hm hi+1Li , where i>1, we can appeal to Case 1 −1 to get h1 ···hmg k ∈ H1 ···HmK. Thus −1 h1 ···hmk = h1 ···hmg k r ∈ H1 ···HmK, as desired.

Corollary 11.3.7 Let H1,...,Hm be ﬁnitely generated subgroups of a free-by-C abstract group R endowed with the pro-C topology, where C is an extension-closed pseudovariety of ﬁnite groups. Then

Cl(H1 ···Hm) = Cl(H1) ···Cl(Hm).

Proof By Corollary 11.3.4 each Cl(Hi) is ﬁnitely generated. It then follows from Theorem 11.3.6 that Cl(H1) ···Cl(Hm) is closed in the pro-C topology of R.The result follows since H1 ···Hm is dense in Cl(H1) ···Cl(Hm).

Putting together Theorems 11.2.2 and 11.3.6 we deduce

Theorem 11.3.8 Let H1,...,Hm be finitely generated subgroups of a free-by-finite abstract group R. Then the subset H1 ···Hm is closed in the profinite topology of R.

11.4 Separability Properties of Polycyclic Groups

In this section we collect some results on polycyclic-by-ﬁnite groups that we shall use later. The methods used to prove these results are very different from the geometric methods that we have developed in this book. However we include them 11.4 Separability Properties of Polycyclic Groups 343 here for completeness and to facilitate the understanding of some theorems later on in this book. We shall prove only some of them here; for the rest we refer to either the original papers or the comprehensive monograph of Dan Segal on polycyclic groups (cf. Segal 1983). An abstract group R is called polycyclic if it admits a series of subgroups

1 = R0 ≤ R1 ≤···≤Rn−1 ≤ Rn = R (11.1) with Ri Ri+1 and Ri+1/Ri is a cyclic group for each i. Polycyclic groups are obviously solvable, but not necessarily nilpotent (e.g., the symmetric group S3). On the other hand, finite solvable groups are polycyclic, and abelian groups are polycyclic if and only if they are finitely generated. Finitely generated nilpotent groups are polycyclic. An abstract group R is polycyclic-by-finite (respectively, nilpotent-by-finite,etc.) if it contains a subgroup of finite index which is polycyclic (respectively, nilpotent, etc.). Observe that a polycyclic-by-finite group is just a virtually polycyclic group (see Sect. 8.2 for this terminology). Similarly the concept of nilpotent-by-finite, etc. is the same as that of virtually nilpotent group, etc. If R is a polycyclic-by-finite group, let h(R) denote its Hirsch number, the number of infinite cyclic quotients in a series (11.1)(h(R) is clearly an invariant of R: it is independent of the chosen series (11.1)). Note that if S R, then h(R) = h(S) + h(R/S).

Theorem 11.4.1 A polycyclic-by-ﬁnite group R is subgroup separable, and a fortiori residually ﬁnite.

Proof Note that every subgroup H of R is finitely generated. Assume that R is infinite; then R contains an infinite normal subgroup T of finite index such that T has a series

1 = L0 L1 ···Ln = T with all quotients Li+1/Li infinite cyclic (cf. Segal 1983, Chap. 1, Proposition 2). Since T is infinite, torsion-free and solvable, the last nontrivial term of its derived series is a characteristic subgroup A of T which is finitely generated, infinite, torsion- free and abelian, and hence A is free abelian and a normal subgroup of R. Let H be a subgroup of R. To prove the proposition we proceed by induction on the Hirsch number h(R) of R.Ifh(R) = 0, the group R is finite and there is nothing to prove. Assume that h(R) ≥ 1 and that the result holds for polycyclic-by-finite groups whose Hirsch number is smaller than h(R). Since R is infinite, it contains a normal nontrivial free abelian subgroup A, as we have pointed out above. If m is a positive integer, let Am be the subgroup of A consisting of all the elements am (a ∈ A). Clearly Am R, and so h(R/Am) = h(R) − h(Am)

Hence M ≤ HAm.

H ≤M≤f R m Define K = HAm. Then, to prove that H = M, it suffices to show m H ≤M≤f R that H = K.Now,A ∩ HAm = (A ∩ H)Am. Since A/A ∩ H is finitely generated abelian, one has 1 = (A/A ∩ H)m = (A ∩ H)Am /(A ∩ H), m m ∩ = ∩ m ∩ = ∩ m i.e., A H m(A H)A . On the other hand, A K m(A H)A . Thus, A ∩ K = A ∩ H . Since H ≤ K ≤ HA, intersecting with K we have H ≤ K ≤ HA∩ K = H(A∩ K) = H ,soK = H , as desired.

Since a ﬁnitely generated nilpotent group is polycyclic, we deduce

Corollary 11.4.2 Finitely generated nilpotent-by-ﬁnite abstract groups are subgroup separable.

Corollary 11.4.3 (a) Let R be a polycyclic-by-finite group and let H be a subgroup of R. Then the profinite topology of H coincides with the topology on H induced by the profinite topology of R. (b) The completion functor on the category of polycyclic-by-finite groups is exact, i.e., if 1 −→ K −→ R −→ H −→ 1 is an exact sequence of polycyclic-by-finite groups, the corresponding sequence of profinite completions 1 −→ Kˆ −→ Rˆ −→ Hˆ −→ 1 is also exact.

Proof Part (a) is a consequence of Theorems 11.4.1 and Lemma 11.1.4. Part (b) follows from part (a) and standard facts about completions (see Proposition 3.2.5 and Lemma 3.2.6 in RZ).

Theorem 11.4.4 (Remeslennikov 1969; Formanek 1970, 1976) Polycyclic-by- ﬁnite groups are conjugacy separable.

Theorem 11.4.5 (Grunewald and Segal 1978) Polycyclic-by-ﬁnite groups are subgroup conjugacy separable.

For a group G and a natural number m, denote by Gm the subgroup gm | g ∈ G of G generated by the m-th powers of its elements. Note that Gm G. 11.4 Separability Properties of Polycyclic Groups 345

Proposition 11.4.6 Let R be a polycyclic-by-ﬁnite group. Then every cyclic subgroup of R is conjugacy distinguished.

ˆ Proof Let x,y be elements of R, and suppose that yR ∩ x=∅. We need to show that yR ∩ x=∅. We shall prove this by induction on the Hirsch number h(R) of R.Ifh(R) = 0, then R is finite and the result is obvious. Say h(R) ≥ 1. Note that if either the order of x or the order of y is finite and ˆ yR ∩ x=∅, then both x and y have finite order and the result is a consequence of Theorem 11.4.4. So, we assume from now on that both x and y have infinite order. Let A be a nontrivial free abelian normal subgroup of R (cf. Segal 1983, Chap. 1, Lemma 6). Then R/A is polycyclic-by-finite with h(R/A) < h(R).Letm m be a natural number, and let πm: R −→ R/A be the canonical epimorphism. By Corollary 11.4.3, R/Am = R/ˆ Am. Consider the commutative diagram π R m R/Am

ι ιm π R m R/Am = R/ˆ Am

Rˆ where the maps ι and ιm are the canonical injections. Note that π m(y ) = R/Am ˆ m (πm(y)) and π m( x) = πm(x); therefore, in R/A we have ˆ m ˆ ˆ m R/A m R R yA ∩ xA = π m y ∩ π m x ≥ π m y ∩ x = ∅. By the induction hypothesis, for each natural number t, there exist some r(t) ∈ R and n(t) ∈ Z such that yr(t) ≡ xn(t) (mod At ). (11.2) Since a conjugate of y in R is an element of x if and only a conjugate of yr(1) in R is an element of x, we may replace y with yr(1), without loss of generality; so we may assume that y ≡ xn(1) (mod A). (11.3) Then for any natural number t one has

xn(t) ≡ yr(t) ≡ xn(1) r(t) (mod A), where r(1) = 1; hence xn(t) and xn(1) have the same order in R/A, and a fortiori in any ﬁnite quotient of R/A. It follows that xn(t) N = xn(1) N whenever A ≤ N f R. By Theorem 11.4.1, R/A is subgroup separable, hence according to Lemma 11.1.4(b), xn(t) A = xn(1) A = yA, for all natural numbers t. Now we consider two cases. 346 11 Separability Conditions in Free and Polycyclic Groups

Case 1. The element yA of R/A has infinite order. Then, so does xA. Hence n(1) =±n(t), and in particular, n(1) =±n(t!), for all natural numbers t.Ifn(1) = n(t!) for infinitely many t, put k = n(1); otherwise, put k =−n(1). Then, according to (11.2), xk and y are conjugate modulo At! for infinitely many t, say, for t = t1,t2,....LetN f R; then there exists some natural m number m such that A ≤ A∩N; since the sequence t = t1,t2,...is infinite, m

Case 2. The element yA of R/A has finite order. Say the order of yA is f ; then xA must also have finite order, say, e.From(11.3) one obtains that e = flfor some natural number l. Since 1 = yf ∈ A and A is a free abelian group, there is some basis {a1,a2,...} of A and some natural number t such t = f t = f × tk that a1 y .SoA y C for some subgroup C of A. Then yA has order fkin the group R/Atk, for every natural number k. Similarly, there exists a natural number s such that xAsk has order ek in R/Ask, for each natural number k.Nextlet w be any common multiple of t and s.From(11.2), the order of xn(w)Aw in R/Aw is then fw/t, while the order of xAw is ew/s = lf w/s . Therefore tl = sq for some q ∈ N, and the order of xq Aw is fw/t. Since the cyclic subgroup xAw of R/Aw has a unique subgroup of order fw/t, we see that xn(w)Aw = xq Aw for all w as above. Therefore, according to (11.2), the groups xq Aw and yAw are conjugate in R/Aw, for all such w.LetN be a normal subgroup of R of finite index; then there exists some natural number w0 such that whenever w0 | w, one has Aw ≤ N; so there exists a w which is a multiple of t and s such that Aw ≤ N.It follows that the groups xq N and yN are conjugate in R/N, for all N f R. Hence xq and y are conjugate in R, since R is subgroup conjugacy separable (Theorem 11.4.5); thus yR ∩ x=∅, as desired.

Proposition 11.4.7 Let R be a polycyclic-by-ﬁnite group and let A and B be subgroups of R. Then the set AB is closed in the proﬁnite topology of R.

Proof According to a theorem of L. Auslander (cf. Auslander 1969; see also Swan 1967 or Segal 1983, Chap. 5, Theorem 5), we may think of R as a subgroup of GLn(Z), for some natural number n.LetM denote the additive group of the ring 11.4 Separability Properties of Polycyclic Groups 347 of n × n matrices Mn(Z) over Z. Then R ≤ GLn(Z) ⊆ M. Consider the group monomorphisms ρ : R −→ Aut(M) and λ : R −→ Aut(M), written r → ρr and r → λr , respectively, (r ∈ R), where if r ∈ R and m ∈ M, −1 then ρr (m) = mr , and λr (m) = rm.ForT ≤ R, deﬁne ρT ={ρt | t ∈ T } and (λ ,ρ ) λT ={λt | t ∈ T }. Observe that if A,B ≤ R, then λA × ρB acts on M by a b m = amb−1 (a ∈ A,b ∈ B,m ∈ M).

Step 1. If T ≤ R, then T is a closed subset in the profinite topology of M. = Consider the semidirect product SρT M ρT ; note that the group SρT is polycyclic-by-finite since M is finitely generated abelian and ρT is polycyclic- by-finite. Hence SρT is conjugacy separable (see Theorem 11.4.4); consequently (see Lemma 11.1.5), every conjugacy class of an element of SρT is closed in the profinite topology of SρT . Now, the conjugacy class in SρT of the identity −1 matrix I is the set T : indeed, if t ∈ T and m ∈ M, (m, ρt ) (I, 1)(m, ρt ) = − = = ( mt, ρt−1 )(I, 1)(m, ρt ) (It, 1) (t, 1). Therefore T is closed in the profinite topology of SρT . Next observe that the profinite topology of SρT induces on M its (full) profinite topology (see Lemma 11.1.3(b)). Since T ⊆ M, we deduce that T is closed in the profinite topology of M.

Step 2. The profinite topology of M induces on R its (full) profinite topology. Let T be a subgroup of R of finite index. We must prove that T is open in the topology of R induced by the profinite topology of M. Observe that this induced topology makes R into a topological group: indeed, first note that the subgroups of M of the form Mn(tZ)(t∈ N) form a fundamental system of neighbourhoods of the zero matrix; hence the profinite topology of M induces on GLn(Z) the congruence subgroup topology (this is in fact the definition of this topology), making it into a topological group; since R is a subgroup of GLn(Z), the assertion follows. According to Step 1, T is closed in the profinite topology of M, and so in the induced topology on R. Since T has finite index in R, it is also open in the induced topology.

Step 3. AB is closed in the profinite topology of R. By Step 2, it suffices to show that AB is closed in the profinite topology of M. Con- sider the semidirect product S = M (λA × ρB ). Now we proceed as in Step 1: first one easily checks that the conjugacy class in S of the identity matrix I is precisely AB; then since S is polycyclic-by-finite, it follows that AB is closed in the profinite topology of S, and so of M. Chapter 12 Algorithms in Abstract Free Groups and Monoids

Let Φ be an abstract free group of finite rank endowed with its pro-C topology. The first section of this chapter is concerned with algorithms to compute the open subgroups Φ. The second section presents a general framework to describe possible algorithms to compute (i.e., to find a basis for) the closure Cl(H ) of a finitely generated subgroup H of Φ, when H is given by a finite set of generators. The third section collects applications of the results in Chap. 11 and the algorithms in Sect. 12.2 of this chapter to free monoids as well as related topics in finite monoids.

12.1 Algorithms for Subgroups of Finite Index

Let Ψ = Ψ(X) be an abstract free group with a finite basis X ={x1,...,xr } endowed with its pro-C topology. The main aim of this section is to describe an algorithm to compute (i.e., find bases for) the open subgroups of Ψ and the intersection of any two of them. We begin with general subgroups of finite index in Ψ .One question that arises is how to identify a subgroup A of finite index in Ψ . One obvious way is to specify a set of generators for A; however, if g1,...,gn ∈ Ψ ,itis not immediately clear whether the subgroup generated by them, g1,...,gn, has finite index or not (although, certainly, there are algorithms to decide this; see, e.g., Stallings 1983, Remark 7.6). Instead, we start with a different description of a subgroup of finite index that is more suitable for our purposes. We frequently use the symmetric group St = Sym(Ω) of degree t; as usual we think of this group as operating on the set Ω of size t. To give an action of an abstract group B on the set Ω corresponds to giving a homomorphism ρ : B → Sym(Ω): for b ∈ B and ω ∈ Ω, ωb = ωρ(b).

Lemma 12.1.1 Let B be an abstract group, t a natural number and A a subgroup of B. Then the following statements are equivalent: (a) The index of A in B is t.

© Springer International Publishing AG 2017 349 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_12 350 12 Algorithms in Abstract Free Groups and Monoids

−1 ˜ (b) A = ρ (A), where ρ is a homomorphism ρ : B → St = Sym(Ω) from B into the symmetric group of degree t, and A˜ is a subgroup of ρ(B) of index t. −1 ˜ (c) A = ρ (A), where ρ is a homomorphism ρ : B → St = Sym(Ω) from B into the symmetric group of degree t such that ρ(B) is transitive (i.e., Ω is an orbit under the action of ρ(B)) and A˜ is the stabilizer of some point of Ω. −1 ˜ (d) A = ρ (A), where ρ is a homomorphism ρ : B → St = Sym(Ω) from B into the symmetric group of degree t such that ρ(B) has an orbit Ω of size t and A˜ is the stabilizer in ρ(B) of some point of Ω . (e) Let Ω be a set of size t . Then there exists a homomorphism B → Sym(Ω) and an element ω ∈ Ω such that the orbit of ω under the corresponding action of B has size t and A is the stabilizer of ω.

(f) Let Ω be a set of size t and let ω0 be a ﬁxed element of Ω. Then there exists a homomorphism B → Sym(Ω) such that the orbit of ω0 under the corresponding action of B has size t and A is the stabilizer of ω0.

Proof It is easy to check that (d), (e) and (f) are equivalent. Clearly, any of (b), (c) or (d) implies (a). Conversely, assume (a) holds, i.e., that [B : A]=t.LetΩ be the set of right cosets A\B of A in B.Let ρ : B −→ Sym(Ω) be the homomorphism deﬁned as (Ay)ρ(x) = Ayx (x, y ∈ B). Note that − Ker(ρ) = y 1Ay, y∈B ˜ −1 ˜ ˜ is the core AB of A in B. Deﬁne A = ρ(A). Hence ρ (A) = A, [ρ(B) : A]=t, Ω is an orbit of ρ(B) and A˜ is the stabilizer of A1 ∈ Ω. Therefore, (a) implies (b), (c) and (d).

Remark 12.1.2 In case (c) of the lemma above one has Ker(ρ) = AB , the core of A in B.

We are particularly interested in applying the above lemma to the case when B is an abstract free group Ψ = Ψ(X)with a ﬁnite basis X ={x1,...,xr }. Given an r-tuple σ = (σ1,...,σr ) of permutations in St , denote by ρσ the unique homomorphism from Ψ to St such that ρσ (xi) = σi (i = 1,...,r). All homomorphisms from Ψ to St have this form. Let Ψ = Ψ(X)be as above and let Ω be a ﬁnite set of size t (a natural number). We refer to a pair of the form (ρ, ω), where ρ : Ψ → Sym(Ω) is a homomorphism and ω ∈ Ω,asanadmissible pair of type t. We say that a subgroup A of Ψ is represented by an admissible pair (ρ, ω) of type t if A is the stabilizer of ω under the action of Ψ on Ω determined by ρ. As an immediate consequence of Lemma 12.1.1,wehave

Corollary 12.1.3 Let Ψ = Ψ(X)be an abstract free group with a ﬁnite basis X = {x1,...,xr }. Let t be a natural number, Ω a set of size t, and ω aﬁxed(but arbitrary) element of Ω. 12.1 Algorithms for Subgroups of Finite Index 351

(a) Let A be a subgroup of Ψ . Then [Ψ : A]≤t if and only if A can be represented by an admissible pair (ρ, ω) of type t. (b) Let A be a subgroup of Ψ of index t , where t ≤ t. Assume that A is represented by an admissible pair (ρ, ω) of type t. Put A˜ = ρ(A). Then A = ρ−1(A)˜ , and

t =[Ψ : A]= ρ(Ψ): A˜ = Orb(ω), where Orb(ω) denotes the orbit of ω under the action of Ψ (or equivalently, under the action of ρ(Ψ)). In particular, if A has index t in Ψ , then [Ψ : A]= |Ω|.

If A has index t in Ψ and it is given explicitly by a finite set of generators A = a1,...,an, then one can find an admissible pair representing A in the following way: choose a fixed (but arbitrary) element ω ∈ Ω, where |Ω|=t; for each r-tuple σ = (σ1,...,σr ) of permutations in St = Sym(Ω), check whether σ1,...,σr is transitive and whether ρσ (A) is the stabilizer of ω in σ1,...,σr . If that is the case, then (ρσ ,ω)is an admissible pair of type t representing A.

Basic Algorithm 12.1.4 (Computes a basis for a subgroup A of index at most t in the free group Ψ = Ψ(X)) Say A is represented by an admissible pair (ρ, ω) of type t. We describe two methods to ﬁnd the index and a basis for A.

Method 1. Compute the stabilizer A˜ of ω in R = ρ(Ψ) and the orbit Orb(ω) of ω under the action of R = ρ(Ψ). Then [Ψ : A]=|Orb(ω)|=[ρ(Ψ) : A˜]. Construct a ˜ ˜ ˜ ˜ ˜ ˜ ˜ ﬁnite graph Γ as follows: V(Γ)= A\R, E(Γ)= A\R × X, with d0(Ag, x) = Ag ˜ ˜ and d1(Ag, x) = Agρ(x) (g ∈ R,x ∈ X). Consider the Cayley graph Γ(Ψ,X); then the graphs A\Γ(Ψ,X)and Γ˜ are isomorphic by means of the map

Af → Aρ(f˜ ), (Af, x) → Aρ(f˜ ), x . Therefore, according to Corollary A.2.8, one has an explicit isomorphism A =∼ abs ˜ abs ˜ π1 (Γ); thus it sufﬁces to obtain algorithmically a basis for π1 (Γ). This is done in Algorithm A.1.6.

Method 2. (More explicit) Recall that a Schreier system in an abstract free group = = %1 ··· %i ··· %s ∪ −1 Ψ Ψ(X) is a set of reduced words w x1 xi xs in X X such that %1 ··· %i every initial segment x1 xi of a word w in the set, is also in the set. We shall compute a Schreier right transversal R of A in Ψ , i.e., a right transversal which is also a Schreier system. We do this using the action of Ψ on Ω determined by ρ. Note that {1} is a Schreier system and 1 represents the vertex A1. Assume R is a Schreier system whose elements represent the different vertices of a subtree T of A\Γ(Ψ,X)containing the vertex H1. Observe that if w1 and w2 are different words in R , then ωρ(w1) = ωρ(w2), because A is the stabilizer of ω. Check whether there is a pair (w, x), with w ∈ R and x ∈ X, such that ωρ(wx) ∈{/ ωρ(w) | w ∈ R }. If no such pair (w, x) exists, then T = T isamaximaltreeofA\Γ(Ψ,X), and R = R is the desired Schreier transversal. If such a pair exists, then T = T ∪{Awx,(Aw,x)} 352 12 Algorithms in Abstract Free Groups and Monoids is a subtree of A\Γ(Ψ,X) strictly larger than T . Deﬁne R = R ∪{wx}. Since A\Γ(Ψ,X)has at most t vertices, by induction we obtain a Schreier right transversal R of A in Ψ and a maximal subtree T . Observe that if w ∈ R and x ∈ X, then there is a unique element in R, denoted wx, with Awx = Awx: it is the unique element wx ∈ R such that ωρ(wx) = ωρ(wx). Then it is well-known that the set − wxwx 1 = 1 w ∈ R,x ∈ X is a basis for A (see, for example, Serre 1980, Sect. I.3.3, Theorem 4 or Kargapolov and Merzljakov 1979, Theorem 14.3.5). Note that [Ψ : A]=|R|.

Next we provide a method to decide whether a subgroup A of Ψ represented by a pair (ρ, ω) of type t is open or not in the pro-C topology of Ψ . The underlying assumption here is that there exists an algorithm that decides whether or not a given finite group of permutations is in C. For example, this is obviously the case if C is the pseudovariety of all finite p-groups, where p is a fixed prime number: in this case one simply has to check whether the orders of the permutations in the subgroup are p-powers.

Basic Algorithm 12.1.5 (Decides openness of subgroups of ﬁnite index in the pro-C topology of the abstract free group Ψ = Ψ(X)) Assume A is a subgroup of index t in Ψ represented by a pair (ρ, ω) of type t. Recall that A is open in the pro-C topology of Ψ if and only if Ψ/AΨ ∈ C (see Lemma 11.1.2). Since Ker(ρ) = AΨ (see Remark 12.1.2), we have the following algorithm: A is open in the pro-C topology of Ψ if and only if ρ(Ψ)∈ C.

Basic Algorithm 12.1.6 (Decides whether a given element of Ψ = Ψ(X) is in a given subgroup of ﬁnite index) Let the subgroup A of Ψ be represented by a pair (ρ, ω) of type t.Letg ∈ Ψ = Ψ(X)be given as a word in X ∪ X−1. Then g ∈ A if and only if ωρ(g) = ω.

Basic Algorithm 12.1.7 (Finds a basis and the index for the intersection of two subgroups of finite index in Ψ = Ψ(X))LetAi be a subgroup of finite index ti in Ψ(i= 1, 2). Say Ai is represented by (ρi,ωi) of type ti (i = 1, 2).LetA = A1 ∩A2. We describe an algorithm to find a basis and the index of A1 ∩ A2 in Ψ . Define a homomorphism

ρ : Ψ −→ Sym(Ω1) × Sym(Ω2) ≤ Sym(Ω1 × Ω2) by ρ(g)= (ρ1(g), ρ2(g)) (g ∈ Ψ). Let A be the stabilizer of ω = (ω1,ω2) under the action of Ψ on Ω1 ×Ω2 induced by ρ. Then

A = A1 ∩ A2 and it is represented by the pair (ρ, ω) of type t1t2. Now one can apply Algo- rithm 12.1.4 to ﬁnd the index and a basis for A = A1 ∩ A2. 12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups 353

Let t be a fixed natural number. Since the free group Ψ has finite rank, it has finitely many subgroups of index at most t, and hence, finitely many open subgroups of index at most t. The following corollary now follows readily from the above algorithms.

Corollary 12.1.8 Let C be a pseudovariety of finite groups. Let Ψ = Ψ(X) be a free abstract group with finite basis X endowed with its pro-C topology. Let H = h1,...,hr be a finitely generated subgroup of Ψ , where each generator hi is given as a word in X ∪ X−1. Let t be a natural number and let U(t) be the intersection of all open subgroups of Ψ of index at most t containing H . Then (a) there exists an algorithm to find a basis for U(t); (b) there exists an algorithm to find the index of U(t) in Ψ ; (c) if g ∈ Ψ , there exists an algorithm to decide whether or not g ∈ U(t).

12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups

In this section the pseudovariety C is assumed to be extension-closed.

Let Φ = Φ(X) be an abstract free group with basis X and let H = h1,...,hr be a finitely generated subgroup of Φ. Then H can be explicitly described by specifying each of the elements hi as a finite product of elements of X and their inverses. As shown in Proposition 11.3.1, the topological closure Cl(H ) of H in the pro-C topology of Φ is also finitely generated (for C an extension-closed pseudovariety of finite groups). In this section we explore the possibility of describing explicitly a set of generators for Cl(H ) (for some class C), i.e., describing an algorithm to find generators for Cl(H ). We shall show that this can be done when C is the class of all finite p-groups, where p is a fixed prime number. We begin with basic considerations valid for a general extension-closed pseudovariety of finite groups C. The words that describe the hi involve finitely many elements of X.SoH is contained in a finitely generated free factor of Φ. Since such a free factor is closed in the pro-C topology of Φ (see Lemma 11.1.3), we may replace Φ with that free factor; hence we will assume from now on that X is finite. Since Φ is finitely generated, it has only finitely many subgroups of a given finite index. For a natural number m,letU(m)denote the intersection of all open (in the pro-C topology of Φ) subgroups of Φ of index at most m containing H ; then U(m) is open. Note that H ≤ Cl(H ) ≤ U(m). Clearly Cl(H ) = U(m). m ∈{ } For each i 1,...,r , denote by p1,hi the unique reduced path from 1 to hi in the abstract Cayley graph Γ = Γ(Φ,X)of Φ with respect to X.Let[1,hi] be the 354 12 Algorithms in Abstract Free Groups and Monoids underlying chain (see Sect. A.1 in Appendix A) determined by 1 and hi . Consider the subtree r S = [1,hi] i=1 of the tree Γ . For a subgroup K such that H ≤ K ≤ Φ, consider the quotient graph K\Γ of Γ under the action of K: its set of vertices is the set of right cosets K\Φ, and its edges have the form (Hg, x) with d0(Hg, x) = Hg and d1(Hg, x) = Hgx (g ∈ Φ,x ∈ X). For example, if H = Φ, Φ\Γ is a bouquet of |X| loops. Let SK denote the image of S in the quotient graph K\Γ ; note that SK is a finite subgraph of K\Γ and one of its vertices is the coset K1. Furthermore, the \ image p1,hi (K) in K Γ of the path p1,hi is a closed path in the graph SK , and so it abs = abs represents an element of the abstract fundamental group π1 (SK ) π1 (SK ,K1) abs \ = abs \ of SK . We always identify K with π1 (K Γ) π1 (K Γ,K1) by means of the natural isomorphism : −→ abs \ = abs \ ρK K π1 (K Γ) π1 (K Γ,K1) defined in Corollary A.2.8; therefore, using part (b) of that corollary and the fact that H is generated by h1,...,hr ,wehave = abs = abs \ ≤ abs ≤ abs \ = ≤ H π1 (SH ) π1 (H Γ) π1 (SK ) π1 (K Γ) K Φ. abs = abs \ By Corollary A.1.5, π1 (SK ) is a free factor of K π1 (K Γ). Assume now that K is open in the pro-C topology of Φ; then the induced topology on K coincides C abs with the full pro- topology of K (see Lemma 11.1.3(c)). Therefore, π1 (SK ) is closed in the pro-C topology of K (see Lemma 11.1.2(d)), and so of Φ (since K is abs closed in Φ, because it is open). Hence, if K is open and H is dense in π1 (SK ), abs = then π1 (SK ) Cl(H ). Consider the diagram τ Γ H\Γ τ Cl(H )\Γ m U(m)\Γ

S SH SCl(H ) SU(m) where the horizontal maps are the natural epimorphisms of graphs. Since SCl(H ) is ﬁnite and Cl(H ) is the intersection of the U(m), there exists some natural number m0 such that the morphism τm is injective on SCl(H ) for m ≥ m0 (see Lemma 8.1.1), so that its restriction to SCl(H ) deﬁnes an isomorphism of the graphs : −→ τm|SCl(H) SCl(H ) SU(m) ≥ ≥ abs abs (m m0). Therefore, for m m0, we can identify π1 (SCl(H )) with π1 (SU(m)). abs = abs Since U(m) is open, the argument above shows that π1 (SCl(H )) π1 (SU(m)) is a closed subgroup of Φ. Since

≤ abs = abs ≤ abs \ = H π1 (SCl(H )) π1 (SU(m)) π1 Cl(H ) Γ Cl(H ), abs = abs = ≥ we deduce that π1 (SU(m)) π1 (SCl(H )) Cl(H ),ifm m0. 12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups 355

Note also that since S is a finite graph, the number of different quotient graphs of abs S is finite; hence, there are only finitely many possible values for π1 (SCl(H )), and so only finitely many possible closures of H for different extension-closed pseudovarieties of finite groups C. We collect all this in the following theorem.

Theorem 12.2.1 Let C be an extension-closed pseudovariety of finite groups. Let H = h1,...,hr be a finitely generated subgroup of an abstract free group Φ = = r [ ] Φ(X) with finite basis X. Let S i=1 1,hi be the corresponding finite subgraph of the Cayley graph Γ = Γ(Φ,X)determined by the elements h1,...,hr . Let SU(m) denote the image of S on the quotient graph U(m)\Γ , where U(m) is as described above. Then

(a) there exists some natural number m0 such that the closure Cl(H ) of H in the pro-C topology of Φ is = abs ≥ ; Cl(H ) π1 (SU(m))(m m0) abs (b) to find such m0 it suffices to verify that H is dense in π1 (SU(m0)) endowed with its own pro-C topology; (c) the number of different closures ClC(H ) of H in Φ in all possible pro-C topologies is finite. In particular, there are only finitely many pro-p closures (i.e., closure in the pro-p topology)Clp(H ) of H in Φ, for all possible prime numbers p.

This theorem suggests a possible algorithm to compute Cl(H ), the closure of H in the pro-C topology of Φ. One simply has to make sure that the steps involved in the theorem can be carried out algorithmically: a procedure to obtain the open (in the pro-C topology) subgroups of index at most m of free abstract groups of finite rank, where each of these free groups is described by a basis (in our case, these abs are the groups π1 (SU(m))); the existence of an algorithm to obtain the intersection U(m) of those open subgroups; and finally, an algorithm to decide whether or not a finitely generated subgroup of a free abstract group is dense (with respect to the C abs pro- topology) in the free group (in our case, whether H is dense in π1 (SU(m))). Algorithms for some of these steps are described in Sect. 12.1. One then easily deduces from this theorem that the crucial difficulty when devising an algorithm to compute Cl(H ) is the following: Let Ψ be a free abstract group with finite basis Y endowed with its pro-C topology. Let H = h1,...,hr be a finitely generated subgroup whose gen- −1 erators hi are specified as words in Y ∪ Y . Is there an algorithm that will decide whether or not H is dense in Ψ ? Of course, the existence of such an algorithm will depend on the extension-closed pseudovariety of finite groups C.

abs Remark 12.2.2 Observe that in our case the groups π1 (SU(m)) play the role of Ψ , and so Ψ and the basis Y changes with m. The subgroup H as well as its generators h1,...,hr are fixed, but the word representing each generator changes with each 356 12 Algorithms in Abstract Free Groups and Monoids new basis. We indicate how to construct these bases and words algorithmically in our case, knowing that U(m) can be described by a basis (see Corollary 12.1.8(a)). Construct SU(m); this can be done algorithmically since one can decide whether two cosets of U(m), determined by two vertices of S, are equal or not according to Corollary 12.1.8(c). abs Then one finds a basis of π1 (SU(m)) following the procedure indicated in Al- gorithm A.1.6: one chooses a maximal tree Tm of the finite subgraph SU(m) of U(n)\Γ ; for an edge e ∈ SU(m), construct a cycle e¯ in SU(m) based at U(m)1by joining the vertices of e to the vertex U(m)1ofSU(m) using the unique paths in Tm joining U(m)1 with those vertices; then

Xm ={¯e | e ∈ SU(m) − Tm} abs ∪ −1 is a basis for π1 (SU(m)). To write the generators of H as words in Xm Xm we proceed as in the proof of Proposition A.1.3:letp1,hi (U(m)) be the image in SU(m) of the path p1,hi ;sayp1,hi (U(m)) (a cycle in SU(m) based at U(m)1) has the form = %1 %s p1,hi (U(m)) e1 ,...,es , where the ei are edges of SU(m). Then the element in abs ¯%1 ¯%s π1 (SU(m)) represented by p1,hi (U(m)) can also be represented by e1 ,...,es in terms of the basis Xm (note that e¯i represents 1 if e ∈ Tm).

Next we describe an algorithm that computes the closure Cl(H ) of H in the pro- p topology of Φ, i.e., when C consists of all finite p-groups, where p is a fixed prime number. The key ideas in this case are the following: if Ψ is an abstract free group of finite rank with the pro-p topology, then (i) every proper open subgroup U of Ψ is contained in an open normal subgroup ˜ U of index p: indeed (see Lemma 11.1.2(a)), since U is open, Ψ/UΨ is a finite p-group, where UΨ denotes the core of U in Ψ ; so there is a subnormal series ˜ UΨ = U0 < ···

Lemma 12.2.3 Let Ψ be a free abstract group of rank n and let H = h1,...,hr be a ﬁnitely generated subgroup of Ψ . Denote by n ρ : Ψ → Z/pZ ⊕ ···⊕ Z/pZ the natural epimorphism, where p is a prime number. Let M be the r × n matrix with integer mod p coefﬁcients, whose i-row is ρ(hi). Then H is dense in the pro-p topology of Ψ if and only if M has rank n, i.e., if and only if ρ(H)= (Z/pZ)n. 12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups 357

Algorithm 12.2.4 (Computes the closure of a finitely generated subgroup of a free abstract group endowed with the pro-p topology) Data: A prime number p. A free abstract group Φ (endowed with the pro-p topology) specified by a finite basis X. A finitely generated subgroup H of Φ, specified by a finite set of genera- −1 tors h1,...,hr , explicitly represented by words in X ∪ X .LetΓ = Γ(Φ,X),the Cayley graph of Φ with respect to X. = r [ ] := Step 1. Construct the finite subgraph S 1 1,hi of Γ . Set n p. Step 2. Construct U(n), the intersection of all open subgroups of Φ of index at most n containing H (see Corollary 12.1.8).

\ abs Step 3. Construct the image SU(n) of S in U(n) Γ and π1 (SU(n)) (see Re- mark 12.2.2). = Step 4. Construct the path p1,hi (U(n)), the image in SU(n) of the path p1,hi in Γ(i abs 1,...,r). These represent elements of π1 (SU(n)). Check whether the subgroup = abs H p1,h1 (U(n)), . . . , p1,hr (U(n)) is dense in π1 (SU(n)) (see Lemma 12.2.3). = abs := + = abs If it is, then Cl(H ) π1 (SU(n)). Otherwise, set n n 1, Φ π1 (SU(n)) and go to Step 2.

Note that Theorem 12.2.1 guarantees that this algorithm stops after a ﬁnite number of steps, and its output is Cl(H ).

Corollary 12.2.5 Let p be a prime number, Φ a free abstract group and let H be a ﬁnitely generated subgroup of Φ. Then there exists an algorithm to decide whether or not H is closed in the pro-p topology of Φ.

Proof According to Algorithm 12.2.4 one can compute (i.e., find a finite number of generators of) the closure Cl(H ) of H in the pro-p topology of Φ. Then it suffices to decide whether or not Cl(H ) is contained in H ; for this there are well-known algorithms (see Algorithm A.4.3 in Appendix A or Lyndon and Schupp 1977, Propo- sition I.2.21).

Let Nil denote the pseudovariety of all ﬁnite nilpotent groups. Although this class is not extension-closed, one can obtain information about the pronilpotent closure ClNil(H ) of a ﬁnitely generated subgroup H of a free group Φ using Theo- rem 12.2.1.

Lemma 12.2.6 Let Φ be a free abstract group. Then (a) for any subgroup H of Φ, one has ClNil(H ) = Clp(H ), p

where p ranges over all prime numbers, and for any prime number p,Clp(H ) denotes the pro-p closure of H in Φ; 358 12 Algorithms in Abstract Free Groups and Monoids

(b) if H is a finitely generated subgroup of Φ, then there exist finitely many prime numbers p1,...,pt such that t = ; ClNil(H ) Clpi (H ) i=1 and (c) if H is a finitely generated subgroup of Φ, then ClNil(H ) is finitely generated.

Proof (a) For any pseudovariety of finite groups C, the pro-C closure ClC(H ) of H is the intersection of subgroups U of Φ which contain H and are open in the pro-C topology of Φ. Hence ClNil(H ) ≤ Clp(H ), for every prime number p. There- fore it is enough to prove the following claim: for any open subgroup U of Φ in the pronilpotent topology, there exist finitely many prime numbers p1,...,ps = s such that U i=1 Ui , where Ui is a subgroup of Φ open in its pro-pi topology (i = 1,...,s). It suffices to prove a corresponding claim in the quotient group ˜ = Φ Φ/UΦ , where UΦ is the core of U in Φ, namely that there exist finitely many ˜ = s ˜ ˜ ˜ prime numbers p1,...,ps such that U i=1 Ui , where Ui is a subgroup of Φ ˜ ˜ open in its pro-pi topology (i = 1,...,s), and where U is the image of U in Φ. ˜ Note that Φ = Φ/UΦ is a finite nilpotent group (see Lemma 11.1.2 (a)), and so it is the direct product of its Sylow subgroups: ˜ ˜ ˜ Φ = P1 ×···×Ps, ˜ where Pi is a finite pi -group, and p1,...,ps are different primes. For each i, put ˜ ˜ ˇ˜ ˜ ˜ Ui = P1 ···Pi ···PsU ˇ˜ ˜ ˜ (here, Pi indicates that Pi is deleted). Then one needs to check that each Ui is open ˜ s ˜ = ˜ ˜ in the pro-pi topology of Φ and i=1 Ui U. Denote by Up the unique p-Sylow subgroup of U˜ ; then ˜ = ˜ ×···× ˜ ×···× ˜ ; Ui P1 Upi Ps and so the assertions are obvious. (b) This follows from part (a) and Theorem 12.2.1(c). (c) This a consequence of part (b) and Howson’s theorem (see Theorem A.4.4 in Appendix A).

Using this lemma and Algorithm 12.2.4, Margolis, Sapir and Weil (2001)show that if H is a ﬁnitely generated subgroup of a free abstract group Φ, there is an algorithm to compute the pronilpotent closure ClNil(H ) of H in Φ.

Open Question 12.2.7 Is there an algorithm that would compute the closure Cl(H ) of a ﬁnitely generated subgroup H of a free abstract group Φ endowed with its prosolvable topology? 12.3 Algorithms for Monoids 359

Finite supersolvable groups are characterized by the fact their maximal subgroups have prime index (cf. Hall 1959, Corollary 10.5.1 and Theorem 10.5.10). This hints at the possibility that variations of the methods of this section may be used for the prosupersolvable topology of a free abstract group.

Open Question 12.2.8 Is there an algorithm that would compute the closure Cl(H ) of a ﬁnitely generated subgroup H of a free abstract group Φ endowed with its prosupersolvable topology?

12.3 Algorithms for Monoids

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

In this section we include applications of the results obtained in Sects. 11.3 and 12.2 to some aspects of the theory of monoids and the theory of formal languages. For basic facts about monoids and formal languages, we refer the reader one of the standard sources on the subject, e.g., Eilenberg (1974). Recall that a monoid consists of a nonempty set M endowed with an associative binary operation M ×M −→ M, denoted by (m1,m2) → m1m2 (m1,m2 ∈ M), that has a neutral element, denoted by 1: (m1m2)m3 = m1(m2m2) and 1m = m1 = m, for all m1,m2,m3,m ∈ M.Amorphism ϕ : M −→ M from a monoid M to a monoid M is a function such that ϕ(m1m2) = ϕ(m1)ϕ(m2) (m1,m2 ∈ M) and ϕ(1) = 1. If M is a monoid and S,T ⊆ M, we deﬁne ST ={st | s ∈ S,t ∈ T }; S2 = SS and inductively, Sn = Sn−1S (n = 2, 3,...). A subset T of M is a submonoid of M if 1 ∈ T and T 2 = T .IfM is a monoid and X ⊆ M, the submonoid X∗ of M generated by X is the smallest submonoid of M containing X: X∗ ={1}∪X ∪ X2 ∪···. Given a set X,thefree monoid X∗ with base X is the monoid consisting of all formal expressions m1m2 ···mt , where t is a natural number and m1,...,mt ∈ X; if the collection {m1,...,mt } is empty, we denote the corresponding formal expression by 1; to make X∗ into a monoid, one deﬁnes a binary operation in X∗ by concatenation:

··· ··· = ··· ··· (m1m2 mt ) m1m2 mt m1m2 mt m1m2 mt ∈ (m1,m2,...,mt ,m1,m2,...,mt X). The set X is sometimes called an ‘alphabet’ and the elements of X∗ are then called ‘words’ on that alphabet. A subset of X∗ is called a language on that alphabet. Let M be a monoid. The family Rat(M) of rational subsets of M is the smallest family S of subsets of M such that (i) the empty subset and the singleton subsets of M belong to S; (ii) if S and T are in S, then so are S ∪ T and ST ; and (iii) if S is in S, so is the submonoid S∗. 360 12 Algorithms in Abstract Free Groups and Monoids

If L is a rational subset of a monoid M, then L can be expressed in terms of ﬁnitely many singletons using the operations union, product and ∗ ﬁnitely many times. We refer to any such way of expressing L as a rational expression (cf. Eilen- berg 1974, Sect. VII.3).

Lemma 12.3.1 Let ϕ : M −→ N be a morphism of monoids. Then (a) If R is a rational subset of M, then ϕ(R) is a rational subset of N. (b) If ϕ is surjective and R is a rational subset of M, then there exists some rational subset R of N with ϕ(R ) = R.

Proof Let F be the family of all subsets R of M such that ϕ(R) ∈ Rat(N). Clearly F contains the empty subset and all singleton subsets of M; moreover F is closed under taking ﬁnite unions and products and under the star operation. Hence Rat(N) ⊆ F. This proves part (a). Assume now that ϕ is surjective. Let R be the collection of subsets of N of the form ϕ(S),forsomeS ∈ Rat(M). Since ϕ is surjective, all the singleton subsets of N are in R. Moreover, R is closed under taking ﬁnite unions and products and under the star operation. Therefore, Rat(N) ⊆ R. This proves part (b).

Next we recall some basic results that will be used later (for the concepts of automaton and recognizable subset, see Appendix B).

Theorem 12.3.2 (cf. Eilenberg 1974, Sect. VII.5) Let X be a finite set. (a) (Kleene’s Theorem) The rational subsets of X∗ are exactly the subsets of X∗ recognized by finite state automata on the alphabet X. (b) A subset L of the free monoid X∗ is rational if and only if there exists a morphism ϕ : X∗ −→ M into a finite monoid M and a subset P of M such that L = ϕ−1(P ); furthermore, ϕ−1(P ) is explicitly given as a rational expression. (c) If S and T are rational subsets (i.e., recognizable subsets) of X∗, so are S ∩ T and S − T .

In general, for a monoid M, deﬁne inductively sets

Rat0(M) ⊆ Rat1(M) ⊆··· as follows

(i) Rat0(M) is the family of all ﬁnite subsets of M; (ii) for a natural number h ≥ 1, Rath(M) consists of the subsets of M that are ﬁnite unions of subsets of the form ∗ ∗ ··· ∗ x1T1 x2T2 xnTn xn+1, (12.1)

where x1,...,xn+1 ∈ M and where each Ti is in Rath−1(M).

It is clear that Rat(M) = Rath(M), h≥0 12.3 Algorithms for Monoids 361 i.e., every rational subset Z of M is a finite union of subsets of the form (12.1); we refer to such a union as a standard rational expression for Z. Moreover, if Z is given by means of a rational expression, there is an effective procedure to find a standard rational expression for it. We shall agree that to specify a rational subset Z of M is simply to give it as a finite union of subsets of the form (12.1); observe that this involves starting with a finite subset of M (i.e., some element of Rat0(M)), then applying finitely many times the operations S ∪ T , ST and S∗ to obtain certain elements of Rat1(M), etc., until we reach the specified rational expression for Z;in other words, Z can be described in terms of finite data. Note, as an example, that a group R is a monoid, and if H is a subgroup of R generated by a finite subset X, then obviously H = (X ∪ X−1)∗; hence H is a rational subset of R. In fact, the converse also holds:

Theorem 12.3.3 (cf. Anissimow and Seifert 1975; Berstel 1979, Theorem III.2.7) A subgroup H of an abstract group R is a rational subset of R if and only if H is ﬁnitely generated.

Lemma 12.3.4 Let R be an abstract group that is residually C, and let S ⊆ R. Then Cl(S∗) = Cl S.

Proof By Lemma 11.1.1,Cl(S∗) = R ∩ S∗ and Cl( S) = R ∩ S, where, as in- ¯ dicated before, if X ⊆ R, then X denotes the closure of X in R ˆ = lim ∈N R/N C ←− N (here N is the collection of all N R with R/N ∈ C). So it sufﬁces to show that S∗ = S. To see this it is enough to prove that the images of S∗ and S on R/N coincide, for every N ∈ N (see Sect. 1.2). But this is clear since a submonoid of a ﬁnite group is a subgroup.

Theorem 12.3.5 Let R be a free-by-C abstract group endowed with its pro-C topology. Then a subset of R is a closed rational subset of R if and only if it is a ﬁnite union of sets of the form

gH1H2 ···Hn, (12.2) where g ∈ R and where H1,H2,...,Hn are ﬁnitely generated subgroups of R that are closed in the pro-C topology of R.

Proof Let F be the class of subsets of R consisting of finite unions of sets of the form (12.2), where g ∈ R and H1,H2,...,Hn are finitely generated subgroups of R that are closed in the pro-C topology of R. By Theorem 12.3.3 every finitely generated subgroup of R is a rational subset, and hence any set in F is rational; it is also closed by Theorem 11.3.8. Next we must show that a closed rational subset of R is in F.Todothiswe define first a class S to consist of all rational subsets S of R whose closure Cl(S) is in F. We claim that in fact S is the class of all rational subsets of R. Observe that the theorem follows from this claim: indeed, if the claim holds and S is a closed rational subset of R, then S = Cl(S) ∈ F. 362 12 Algorithms in Abstract Free Groups and Monoids

To prove the claim we must show that S satisfies the conditions (i)Ð(iii) above that characterize rational subsets (since S ⊆ Rat(R)). Obviously the empty set and all the singleton sets belong to S, so (i) holds. Let S,T ∈ S; then plainly Cl(S ∪ T)= Cl(S) ∪ Cl(T ) ∈ F;so,S ∪ T ∈ S. To finish the verification of (ii), we must also show that ST ∈ S.Byas- sumption, Cl(S), Cl(T ) ∈ F. So their product Cl(S)Cl(T ) is a finite union of sets of the form g1H1 ···Ht g2K1 ···Kt , with g1,g2 ∈ R and each Hi and Kj a closed finitely generated subgroup of R; since g1H1 ···Ht g2K1 ···Kt = − − 1 ··· 1 ··· ∈ F g1g2(g2 H1g2) (g2 Ht g2)K1 Kt , we deduce that Cl(S)Cl(T ) ; in particular Cl(S)Cl(T ) is closed. Using continuity of multiplication we have ST ⊆ Cl(S)Cl(T ) ⊆ Cl(ST ), and thus Cl(S)Cl(T ) = Cl(ST ). Therefore Cl(ST ) ∈ F, i.e., ST ∈ S. To prove (iii), consider S ∈ S. By Lemma 12.3.4,Cl(S∗) = Cl S. Since S is rational, so is S−1 ={s−1 | s ∈ S}, because for a rational expression describing S, there is a corresponding rational expression to describe S−1; hence S={S ∪S−1}∗ is a rational subset of F. Therefore S is finitely generated, according to Theo- rem 12.3.3. Then, by Corollary 11.3.4,Cl S is also finitely generated; so Cl(S∗) = Cl S∈F, and thus S∗ ∈ S.

Corollary 12.3.6 Let R be a free-by-C abstract group endowed with the pro-C topology. (a) The closure Cl(S) of a rational subset S of R is rational. (b) If L1 and L2 are closed rational subsets of R, then so is their product L1L2. (c) If S1 and S2 are rational subsets of R, then Cl(S1S2) = Cl(S1)Cl(S2).

Next we turn to the question of whether one can ‘compute’ the closure Cl(S) of a given rational subset of a free-by-C group R in its pro-C topology. In the following theorem we see that the computation can be accomplished for certain R and certain classes C.LetS be a rational subset of a free-by-C group R.IfS ∈ Rat0(R), then S is a finite subset of R, and so Cl(S) = S; while, according to Lemma 12.3.4, Cl(S∗) = Cl( S);ifC is the class of all finite groups, then Cl(S∗) = Cl( S) = S (see Theorem 11.2.2), while if R = Φ is a free nonabelian group and C is the class of all finite p-groups, then there is an algorithm (see Algorithm 12.2.4) to compute ∗ a finite set of generators YS for Cl(S ) = Cl( S). Assume next that S ∈ Rath(R), where h ≥ 1, and that whenever T ∈ Rath−1(R), ∗ then Cl(T ) can be computed and a finite set of generators YT for Cl(T ) = Cl T can be computed; then, since S is a finite union of sets the form (12.1), we have that (using Corollary 12.3.6)Cl(S) is a union of sets of the form

x1Cl T1x2Cl T2···xnCl Tnxn+1 = x Y x Y ···x Y x + 1 T1 2 T2 n Tn n 1 −1 ∗ −1 ∗ −1 ∗ = x Y ∪ Y x Y ∪ Y ···x Y ∪ Y x + ; 1 T1 T1 2 T2 T2 n Tn Tn n 1 12.3 Algorithms for Monoids 363 hence Cl(S) is computable as a rational expression; furthermore, one can compute ∗ a ﬁnite set of generators YS for Cl(S ) = Cl S as the union of sets of the form x + x x + x ···x + x x ···x + ,Y n 1 ,Y n n 1 ,...,Y 2 n 1 . 1 2 n 1 Tn Tn−1 T1 Therefore we have proved the following result.

Theorem 12.3.7 Let R be either (a) a free-by-finite abstract group with its profinite topology, or (b) a nonabelian abstract free group with its pro-p topology, for a fixed prime p. Then given a rational subset S of R in terms of a rational expression, there exists an algorithm to compute the closure Cl(S) as a rational expression.

In a free abstract group Φ there is an algorithm to decide whether two rational expressions represent the same rational subset of Φ (cf. Henckell, Margolis, Pin and Rhodes 1991, Lemma 5.11). Therefore one deduces the following consequence.

Corollary 12.3.8 Let S be a rational subset of the free abstract group Φ given by a rational expression. Then it is decidable whether or not S is closed in the pro-p topology of Φ.

If X is a set, then X ⊆ X∗ ⊆ X=Φ(X), where Φ(X) is the abstract free group on X. The pro-C topology of Φ(X) induces on X∗ a topology, that we call the pro-C topology of X∗.IfS ⊆ X∗, we shall denote by Clm(S) the closure of S in the pro-C topology of X∗.

Lemma 12.3.9 Let X be a ﬁnite set. Endow X∗ with the pro-C topology. Let S be a subset of the free monoid X∗. Then (a) Clm(S) = Cl(S) ∩ X∗, where Cl(S) is the closure of S in the pro-C topology of the abstract free group Φ(X) on X; (b) t ∈ Clm(S) if and only if ϕ(t) ∈ ϕ(S) for every morphism ϕ : X∗ −→ G from X∗ into a ﬁnite group G ∈ C.

Proof Part (a) follows from the deﬁnition of the pro-C topology on X∗. Note that the morphisms X∗ −→ G into a group G ∈ C are precisely the restrictions of the homomorphisms Φ(X) −→ G of groups. Now, the elements in Cl(S) are exactly those elements x ∈ Φ(X) such that ϕ(x) ∈ ϕ(S) for every homomorphism ϕ : Φ(X) −→ G into a group G ∈ C (see Sect. 1.2); so part (b) follows from (a).

The above lemma brings to the fore a problem concerning the precise meaning of the computation of an expression. According to Theorem 12.3.5, a closed rational subset of a free abstract group Φ (endowed with its pro-C topology) is a finite union of expressions of the form (12.2). To describe such an expression it suffices to 364 12 Algorithms in Abstract Free Groups and Monoids specify the element g and generators for the subgroups H1,...,Hn of Φ = Φ(X). However, if using Lemma 12.3.9 one wants to ‘compute’ Clm(S) = Cl(S) ∩ X∗,it is clear that one needs more than just the indicated description of Cl(S). To compute Clm(S) one needs an algorithm to decide whether or not a given element of X∗ is in Cl(S). In essence, one requires an algorithm to decide whether or not an element f ∈ Φ is in a product H1 ···Hn of finitely many subgroups Hi of Φ specified by finite sets of generators. The existence of such an algorithm is a consequence of a theorem of Benois (1969) which, for completeness, is included in Appendix B, Theorem B.3.3: it says that H1 ···Hn can be embedded as a recognizable subset of a free monoid on a certain alphabet, and so it is the language recognized by an appropriate automaton on that alphabet.

Theorem 12.3.10 Let X be a finite set and let C be either the class of all finite groups or the class of all finite p-groups for a fixed prime number p. Endow X∗ with the pro-C topology. Let L be a rational subset of the free monoid X∗ given as a rational expression. Then there is an effective procedure to compute the closure Clm(L) of L in the pro-C topology of X∗.

Proof Observe that L is also a rational subset of the free group Φ = Φ(X). Hence, according to Theorem 12.3.7, we can describe Cl(L) as a rational expression. Since Clm(L) = Cl(L) ∩ X∗, the existence of an algorithmic procedure to describe Clm(L) follows from Theorem B.3.3 in Appendix B.

The Kernel of a Finite Monoid

Next we consider a fruitful conjecture raised first by J. Rhodes. The question was posed originally as to whether it is possible to describe algorithmically a certain submonoid, the so-called ‘kernel’ of a finite monoid. We start by describing this ‘kernel’. The identity element of a monoid is always denoted by 1. A relational morphism τ : M N from a monoid M to a monoid N, is a map that assigns to each element m ∈ M a nonempty subset τ(m)of N such that 1 ∈ τ(1) and τ(mm ) ⊇ τ(m)τ(m ), for every m, m ∈ M. For example, if ρ : N −→ M is an epimorphism of monoids, the map m → ρ−1(m) is a relational morphism from M to N. If τ : M N is a relational morphism of monoids and y ∈ N, then we define τ −1(y) = x ∈ M y ∈ τ(x) .

Observe that if y is an idempotent element of N, i.e., if y2 = y, then τ −1(y) is a submonoid of M; in particular, τ −1(1) is a submonoid of M. The proof of the next lemma is straightforward and we leave it to the reader 12.3 Algorithms for Monoids 365

Lemma 12.3.11 Let τ : M N be a relational morphism. Then its graph R = (m, n) ∈ M × N n ∈ τ(m) is a submonoid of M × N. Moreover, for the restrictions α : R −→ M and β : R −→ N of the projections M × N −→ M and M × N −→ N, respectively, one has (a) α is a surjective morphism, (b) τ = βα−1.

Define the C-kernel of a finite monoid M as −1 KC(M) = τ (1), τ where τ ranges over all relational morphisms from M to every group G ∈ C (more precisely, representatives of the isomorphism classes of groups in C). Note that, as − indicated above, τ 1(1) ={x ∈ M | 1 ∈ τ(x)}. Then clearly KC(M) is a submonoid of M.IfC consists of all finite groups (respectively, all finite p-groups, where p is a fixed prime number), then we use the term kernel and notation K(M) (respectively, p-kernel and notation Kp(M)). Then the question of Rhodes can be formulated as follows: Is there an algorithm to compute the kernel K(M) of a finite monoid M? The corresponding pro-p version of this question was posed by J.-E. Pin:

Is there an algorithm to compute the p-kernel Kp(M) of a ﬁnite monoid M?

We give positive answers to these questions below. First, we describe KC(M) in terms of the pro-C topology of a free monoid.

Proposition 12.3.12 Let M be a ﬁnite monoid and let π : X∗ −→ M be a monoid ∗ epimorphism of a free monoid X onto M, where X is a ﬁnite set. Then t ∈ KC(M) if and only if 1 ∈ Clm(π−1(t)), where Clm(π −1(t)) is the closure of π −1(t) in the pro-C topology of X∗.

∗ Proof First assume that t ∈ KC(M) and let ϕ : X −→ G be a monoid morphism from X∗ into a group G ∈ C. Then τ = ϕπ−1 is a relational morphism from M − to G; therefore 1 ∈ τ(t) by the deﬁnition of KC(M). Thus ϕ(1) = 1 ∈ ϕπ 1(t) for any group G ∈ C and any morphism ϕ from X∗ to G. Therefore, 1 ∈ Clm(π −1(t)) (see Lemma 12.3.9). Conversely, assume that t ∈ M and that 1 ∈ Clm(π−1(t)). To verify that t ∈ KC(M), we must show that given a relational morphism τ : M G from M to a group G in C, one has t ∈ τ −1(1), i.e., 1 ∈ τ(t). As in Lemma 12.3.11,let R be the graph of τ ; then τ = βα−1, where α : R −→ M and β : R −→ G are 366 12 Algorithms in Abstract Free Groups and Monoids the projections. By the universal property of free monoids, there exists a morphism γ : X∗ −→ R such that π = αγ . X∗ γ

π R α β

M τ G Now, since 1 ∈ Clm(π −1(t)), it follows from Lemma 12.3.9(b) that 1 = βγ(1) ∈ βγ(π−1(t)). Choose w ∈ π −1(t) with 1 = βγ(w). Then π(w) = t = αγ(w). Set r = γ(w). Then α(r) = t. Since β(r) = 1, we deduce that 1 ∈ βα−1(t) = τ(t),as needed.

Now we can give positive answers to both questions above:

Theorem 12.3.13 Let M be a ﬁnite monoid and let p be a ﬁxed prime number. Then there exist algorithms to compute the kernels K(M) and Kp(M).

Proof Let X be a ﬁnite set, and let π : X∗ −→ M be an epimorphism of monoids. According to Proposition 12.3.12, the computation of KC(M) is equivalent to the computation of Clm(π −1(t)), for any given t ∈ M. Observe that π −1(t) is a rational subset of X∗ and it can be given as an explicit rational expression (see Theo- rem 12.3.2). Hence the result follows from Theorem 12.3.10.

The Mal’cev Product of Pseudovarieties of Monoids

A pseudovariety of finite monoids V is a nonempty collection of (isomorphism classes of) finite monoids which is closed under submonoids, epimorphic images and finite direct products. Observe that a pseudovariety of finite groups is also a pseudovariety of finite monoids. A pseudovariety V of finite monoids is said to be decidable if there is an algorithm to determine whether or not a given monoid is in the pseudovariety. If V and W are pseudovarieties of finite monoids, the Mal’cev product V−1W is the pseudovariety of finite monoids − V 1W = M there is a relational morphism τ : M−→N, with N ∈ W, − such that τ 1(e) ∈ V for all idempotents e ∈ N .

Lemma 12.3.14 Let V be a pseudovariety of ﬁnite monoids and let C be an extension-closed pseudovariety of ﬁnite groups. Then the following conditions are equivalent: (a) M ∈ V−1C; (b) KC(M) ∈ V. 12.3 Algorithms for Monoids 367

Proof (a)⇒(b). Let M ∈ V−1C. Then there exists a relational morphism − τ : M G toagroupG ∈ C such that τ 1(1) ∈ V. Since KC(M) is a sub- − monoid of τ 1(1),KC(M) ∈ V. (b)⇒(a). Since M is ﬁnite there exist groups G1,...,Gn in C and relational morphisms τ1 : M G1 ,..., τn : M Gn such that n = −1 KC(M) τi (1). i=1

Deﬁne τ : M G1 ×···×Gn by τ(m)= (τ1(m),...,τn(m)) (m ∈ M). Then −1 τ is a relational morphism from M to G1 ×···×Gn, and KC(M) = τ (1).It follows that M ∈ V−1C, as required.

Theorem 12.3.15 Let V be a decidable pseudovariety of finite monoids, and let C be either the pseudovariety of all finite groups or the pseudovariety of all finite p-groups, for a fixed prime number p. Then V−1C is decidable.

Proof This follows from Lemma 12.3.14 and Theorem 12.3.13. Chapter 13 Abstract Groups vs Their Proﬁnite Completions

Let R be an abstract group which is residually C. Recall (see Sect. 11.1) that we ¯ denote by X the topological closure of X in the pro-C completion RCˆ of R, while Cl(X) denotes the closure of X in R endowed with its pro-C topology. In this chapter we study the relationship between certain constructions in R and the corresponding constructions in the pro-C completion RCˆ of R; we are particularly interested in intersections of subgroups, normalizers and centralizers: if H1,H2 ≤ R, we want to compare H1 ∩ H2 with H1 ∩ H2;ifH ≤ R, we want to ¯ compare NR(H ) with NRCˆ (H);etc. The main results are obtained when R is an abstract free-by-C group or a polycyclic-by-ﬁnite group (in the latter case we always assume that C is the class of all ﬁnite groups).

13.1 Free-by-Finite Groups vs Their Proﬁnite Completions

Throughout this section C denotes an extension-closed pseudovariety of ﬁnite groups.

We begin with general considerations valid for residually C groups.

Lemma 13.1.1 Let R be an abstract group which is residually C. Let H and K be subgroups of R, which are closed in the pro-C topology of R. Then ∩ ¯ = R NK¯ (H) NK (H ). ¯ ¯ Proof First we claim that NK (H ) = NK (H). Clearly NK (H ) ≤ NK (H). Con- ¯ −1 ¯ versely, let k ∈ NK (H); then if h ∈ H ,wehavek hk ∈ H ∩ R = H , since H is closed in the proﬁnite topology of R (see Lemma 11.1.1(c)); therefore k ∈ NK (H ). This proves the claim. Since K is also closed in the pro-C topology of R,wehave ∩ ¯ = ∩ ¯ = ¯ = R K K. It follows that R NK¯ (H) NK (H) NK (H ).

© Springer International Publishing AG 2017 369 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_13 370 13 Abstract Groups vs Their Proﬁnite Completions

Proposition 13.1.2 (See also Proposition 13.1.4) (a) Let R be a residually C abstract group endowed with its pro-C topology and let H be a subgroup of R. Assume that the pro-C topology of H coincides with the ¯ = topology induced from R, i.e., that H HCˆ. Let U be an open normal subgroup of R. Then U ∩ H = U¯ ∩ H.¯ (b) Let R be a free-by-C abstract group endowed with its pro-C topology. Let H be a ﬁnitely generated closed subgroup and let U be an open normal subgroup of R. Then U ∩ H = U¯ ∩ H.¯

Proof According to Corollary 11.2.5, under the hypotheses of part (b) we have that ¯ ¯ ¯ H = HCˆ. Therefore it sufﬁces to prove part (a). Note that UH = UH = (UH )Cˆ ¯ ¯ (see Lemma 11.1.3(c)) and [R : UH]=[RCˆ : UH ] (cf. RZ, Proposition 3.2.2). So [UH : U]=[U¯ H¯ : U¯ ]. Therefore [H : U ∩ H]=[H¯ : U¯ ∩ H¯ ]. ¯ ¯ Since H = HCˆ, we deduce that if X ⊆ H , then the notation X is unambiguous: it is the closure of X both in HCˆ and in RCˆ. Since U ∩ H is open in H , we can use the above argument to get [H¯ : U ∩ H]=[H : U ∩ H]. Therefore, [H¯ : U ∩ H ]=[H¯ : U¯ ∩ H¯ ]. Since U ∩ H ≤ U¯ ∩ H¯ , one deduces that U ∩ H = U¯ ∩ H¯ .

Lemma 13.1.3 Let H ∈ C be a group of prime order p. Let R = Φ H be a semidirect product, where Φ is an abstract free group. Then there is a free factor Φ1 of Φ such that = × ¯ = × (a) NR(H ) H Φ1 and NRCˆ (H) H (Φ1)Cˆ; and = = (b) CΦ (H ) Φ1 and CΦCˆ (H ) (Φ1)Cˆ. Consequently, = (a ) NR(H ) NRCˆ (H ); = (b ) CΦ (H ) CΦ¯ (H ).

Proof One knows (see Theorem 10.9.3) that the group R is a free product

R = ∗(Ci × Φi) ∗ L, i∈I where L and each Φi are free groups and the Ci are groups of order p.Wemay assume that each Φi is a subgroup of Φ because Ci × Φi = Ci ×[(Ci × Φi) ∩ Φ]. By the Kurosh subgroup theorem for free products of abstract groups, we have that each Φi = (Ci × Φi) ∩ Φ is a free factor of Φ. Therefore, (Φi)Cˆ = Φi , for each i ∈ I (see Lemma 11.1.3(b)). 13.1 Free-by-Finite Groups vs Their Proﬁnite Completions 371

Since any ﬁnite subgroup of R of order p is conjugate to one of the Ci (cf. Magnus, Karrass and Solitar 1966, Corollary 4.1.4), we may assume without loss of = ∈ generality that H Ci1 , for some ﬁxed i1 I . Then = × ∗ = × ∗ R (Ci1 Φi1 ) R1 (H Φ1) R1, = where Φ1 Φi1 and R1 is a subgroup of R. It follows that

NR(H ) = H × Φ1 = CR(H ) (cf. Magnus, Karrass and Solitar 1966, Corollary 4.1.5). Hence

NΦ (H ) = CΦ (H ) = Φ1. Now,

RCˆ = H × (Φ1)Cˆ (R1)Cˆ. Hence (see Corollary 7.1.6) = = × = × CRCˆ (H ) NRCˆ (H ) H (Φ1)Cˆ H Φ1. Finally, using Proposition 13.1.2 we have ¯ = ∩ ¯ = × ∩ ¯ = × ∩ = CΦ (H ) CRCˆ (H ) Φ H Φ1 Φ (H Φ1) Φ Φ1. The following proposition generalizes Proposition 13.1.2.

Proposition 13.1.4 Let R be an abstract free-by-C group, endowed with the pro-C topology. Let H1,H2 be ﬁnitely generated closed subgroups of R. Then

H1 ∩ H2 = H1 ∩ H2 in RCˆ.

Proof Obviously H1 ∩ H2 ≤ H1 ∩ H2; so, it is enough to prove that

H1 ∩ H2 ≤ H1 ∩ H2. (13.1)

Say Φ1 is an open free subgroup of R. By Theorem 11.2.4(b) there exists an open subgroup Φ of Φ1 such that

Φ = (H1 ∩ Φ1) ∗ M = (H1 ∩ Φ)∗ M, for some subgroup M of Φ. Note that the pro-C topology of the clopen subgroup Φ coincides with the topology induced from the pro-C topology of R (see ¯ ¯ Lemma 11.1.3(c)), so that Φ = ΦCˆ ≤ RCˆ; hence, if X ⊆ Φ, the notation X is unambiguous: it represents both the closure in ΦCˆ or in RCˆ. We claim that to prove (13.1) it sufﬁces to prove that

(H1 ∩ Φ)∩ (H2 ∩ Φ) = H1 ∩ H2 ∩ Φ. (13.2) ∩ Indeed, assume that (13.2) holds. Since Hi Φ is open in Hi , one has a ﬁ- = si ∩ (i) (i) (i) ∈ = nite union Hi j=1(Hi Φ)gj , where g1 ,...,gsi Hi ; therefore Hi 372 13 Abstract Groups vs Their Proﬁnite Completions si ∩ (i) = ∈ ∩ = = j=1(Hi Φ)gj (i 1, 2). To prove (13.1), let u H1 H2. Then u a1h1 a2h2 (ai ∈ Hi ∩ Φ, hi ∈ Hi )(i = 1, 2). So, −1 = −1 ∈ ∩ ∩ ∩ a1 a2 h1h2 (H1 Φ)(H2 Φ) R.

Observe that (H1 ∩ Φ)(H2 ∩ Φ) = (H1 ∩ Φ)(H2 ∩ Φ). By Theorem 11.3.6

(H1 ∩ Φ)(H2 ∩ Φ) is a closed subset in the pro-C topology of Φ;so(H1 ∩ Φ)(H2 ∩ Φ) ∩ R = ∩ ∩ −1 = −1 = −1 ∈ ∩ (H1 Φ)(H2 Φ). We deduce that a1 a2 h1h2 b1 b2, where bi Hi Φ (i = 1, 2). Therefore, b1h1 = b2h2 ∈ H1 ∩ H2. Also, using assumption (13.2), = −1 = −1 ∈ ∩ ∩ ∩ = ∩ ∩ v b1a1 b2a2 (H1 R) (H2 R) H1 H2 R. Thus, −1 u = a1h1 = v b1h1 ∈ H1 ∩ H2, proving the claim. It remains to prove (13.2). To simplify the notation we shall restate (13.2)inthe following manner: assume that Φ is a free abstract group, H1 and H2 are closed finitely generated subgroups of Φ and Φ = H1 ∗ M, where M is a subgroup of Φ; then (13.2) says H1 ∩ H2 = H1 ∩ H2. We shall prove this. Since H1 and H2 are finitely generated, they are contained in a finitely generated free factor Ψ of Φ. Since Ψ is closed in the pro-C topology of Φ and its pro-C topology coincides with the topology induced from Φ (see Lemma 11.1.3(b)), we may replace Φ with Ψ if necessary, and so from now on we will assume that Φ is a free group of finite rank. Put H = H1 ∩ H2; note that H is also finitely generated by Howson’s theorem (see Theorem A.4.4 in Appendix A). We need to prove ¯ H = H1 ∩ H2. To do this we embed Φ in an appropriately chosen larger group L which we construct as follows: consider an isomorphic copy Φ of Φ under an isomorphism ρ : Φ −→ Φ ; if a is an element (respectively, a subset) of Φ, we denote by a the corresponding element ρ(a) (respectively, subset) of Φ under that isomorphism; furthermore, we assume that this isomorphism is the identity on H1, i.e., it identifies H1 with H ;so 1 that Φ ∪ Φ is an amalgam of groups with Φ ∩ Φ = H1. Set = ∗ L Φ H1 Φ , the amalgamated free product of the groups Φ and Φ amalgamating H1. Obviously L is a free group of finite rank; in fact

L = M ∗ H1 ∗ M = Φ ∗ M = M ∗ Φ . (13.3) 13.1 Free-by-Finite Groups vs Their Proﬁnite Completions 373

By the Kurosh subgroup theorem for subgroups of free products of abstract groups (see, for example, Serre 1980, Chap. I, Theorem 14) applied to (13.3)we have = ∩ ∗ = ∩ ∗ H2 (H1 H2) A and H2 (H1 H2) A , where A is a subgroup of H2. = ∩ = Observe that the subgroups Φ,Φ ,H1,H2,H2, H H1 H2 H , A and A are all ﬁnitely generated. Furthermore, they are closed in the pro-C topology of L and their pro-C topologies coincide with the topologies induced from the pro-

C topology of L: indeed, Φ, Φ and H1 are free factors of L, and so for these groups the statements follow from Lemmas 11.1.2 and 11.1.3; in the case of H2 and H , we know that these subgroups are closed in R, and so for these groups the statements follow from Corollary 11.2.5; similarly for H ; ﬁnally, A and A 2 are closed since they are free factors of H2 and H2, respectively. In particular, the ¯ notation Φ,H1, H2, etc., is unambiguous: it has the same meaning whether these closures are taken in LCˆ,ΦCˆ, etc.; so from now on, closures are assumed to be taken place in LCˆ, and they coincide with their own pro-C completions: H2 = (H2)Cˆ,etc. Therefore we have ¯ ¯ ¯ ¯ L ˆ = M H M = Φ M = M Φ = Φ Φ = Φ ˆ Φ C 1 H1 C (H1)Cˆ Cˆ (here stands for free pro-C product, and denotes the amalgamated free pro-C H1 product amalgamating H1).

Consider the subgroup P = H2,H of L generated by H2 and H . Then 2 2 = = ∗ = ∗ ∩ ∗ P H2,H2 H2 H1∩H2 H2 A (H1 H2) A . Next we assert that ¯ ∩ ¯ = ¯ ∩ ¯ = P Φ H2 and P Φ H2. ¯ ¯ ¯ To see this define a continuous epimorphism ϕ : LCˆ −→ Φ by sending Φ to Φ iden- ¯ ¯ −1 tically, and Φ to Φ by means of ρ (by definition these maps coincide on H1). ¯ ¯ ¯ Note that ϕ(P)= H2;soP ∩ Φ = H2 follows. The other assertion is proved similarly. We deduce that ¯ ∩ = ¯ ∩ ∩ = ¯ ∩ ¯ ∩ ∩ = ∩ ∩ = ∩ P H1 P H1 H1 P Φ Φ H1 H2 H2 H1 H2 H1 ∩ = ∩ (the last equality holds since by the definition of Φ one has H2 H1 H2 H1). We claim that P is closed in the pro-C topology of L, i.e., that P = L ∩ P.¯ (13.4) To prove this we use the standard tree Sabs associated with the amalgamated free = ∗ C product L Φ H1 Φ , and the standard -tree S associated with the amalga- ¯ mated free pro-C product L ˆ = Φ Φ ; in these cases L = Φ ∗ Φ and C H1 H1 ¯ L ˆ = Φ Φ are the abstract and pro-C fundamental groups of graphs of groups C H1 over a graph with a single edge and two different vertices. We recall the explicit 374 13 Abstract Groups vs Their Profinite Completions definitions of Sabs and S in these specific situations: the vertices of Sabs are the elements of L/Φ ∪. L/Φ and its set of edges is L/H1; moreover, the origin of an edge xH1 (x ∈ L)isd0(xH1) = xΦ, and its terminal vertex is d1(xH1) = xΦ . Similarly, ¯ . ¯ the C-tree S has vertices LCˆ/Φ ∪ LCˆ/Φ and edges LCˆ/H1, with d0(xH1) = xΦ and d1(xH1) = xΦ , where x ∈ LCˆ.Now,themap Sabs −→ S ¯ given by xΦ → xΦ, xΦ → xΦ and xH1 → xH1 (x ∈ L) is an embedding of graphs because by assumption the subgroups Φ, Φ and H1 are closed in the pro-C topology of L (see Sect. 6.5). So we think of Sabs as being a dense subgraph of S. abs abs Denote by e ∈ S ⊆ S the edge e = 1H1 = 1H1, so that S = L{e,d0(e), d1(e)} and S = LCˆ{e,d0(e), d1(e)}. Choose g ∈ L ∩ P¯. To prove (13.4) we need to show that g ∈ P . Note that ge ∈ Pe¯ ⊆ S. Recall that [e,ge] denotes the smallest C-subtree of S containing abs abs e and ge. Since e,ge ∈ S , [e,ge] is a finite subtree of S . Note that H2 fixes = ¯ = ¯ d0(e) 1Φ and H2 fixes d1(e) 1Φ . Since P is generated topologically by H2 and H , and since the segment {e,d0(e), d1(e)} is obviously connected, we deduce ¯2 that P {e,d0(e), d1(e)} is a C-subtree of S (see Lemma 2.2.4 and Proposition 2.4.3); ¯ therefore, [e,ge]⊆P {e,d0(e), d1(e)}.Now,since[e,ge] is finite, it consists of a finite sequence of edges

e,p¯1e,p¯1p¯2e,p¯1p¯2p¯3e,...,p¯1p¯2 ···p ¯ne = ge, ¯ where the elements p¯1, p¯2,...,p¯n belong to Φ or Φ , by the definition of S.Next we assert that the p¯i can be modified in such a way that also p¯i ∈ P . Indeed, since ¯ ¯ [e,ge]⊆P {e,d0(e), d1(e)}, there exists a x1 ∈ P such that x1e =¯p1e;sox1 =¯p1t1, ∈ ¯ −1 ¯ where t1 H1, because H1 is the stabilizer of e; observe that still p1t1 and t1 p2 ¯ −1 are in either Φ or Φ ; hence, replacing p¯1 with p¯1t1 and p¯2 with t p¯2,wemay ¯ 1 assume that p¯1 ∈ P ; the assertion follows inductively. Therefore we have shown ¯ ¯ ∩ ¯ = ¯ ∩ = = that pi is in either P Φ H2 or P Φ H2 (i 1,...,n). Furthermore, since ¯ ¯ p¯1p¯2 ···p ¯ne = ge, we deduce that g =¯p1p¯2 ···p ¯nh, where h ∈ H1. Now, since the C subgroups H1, H2 and H2 are finitely generated and closed in the pro- topology of the free abstract group L, we have that any finite product K1 ···KnH1, with Ki ∈ { } ··· = ∩ ··· H2,H2 , is closed (see Theorem 11.3.6); hence K1 KnH1 L (K1 Kn H1). Since g ∈ L, we deduce that g = p1p2 ···pnh, where the elements p1,p2,...,pn ∈ ∈ ¯ belong to either H2 or H2, and h H1. So, since g P ,wehave ¯ ¯ h ∈ P ∩ H1 = P ∩ H1 ∩ H1 = H2 ∩ H1 ∩ H1 = H2 ∩ H1 = H2 ∩ H1, because H2 and H1 are closed. Thus g ∈ P , as required. This proves the claim. Therefore ¯ = = ∩ = ¯ ∩ P PCˆ ACˆ (H1 H2)Cˆ ACˆ A (H1 H2) A . Also = ∩ ¯ = ∩ H2 (H1 H2) A and H2 (H1 H2) A . 13.1 Free-by-Finite Groups vs Their Profinite Completions 375

We deduce that ∩ = ∩ H2 H2 H1 H2. ∩ = ∩ Thus, since H1 H2 H1 H2, we obtain ∩ = ∩ ∩ ∩ = ∩ H1 H2 H1 H2 H1 H2 H1 H2, as required.

Corollary 13.1.5 Let R be an abstract free-by-C group, endowed with the pro-C topology. Let H be a ﬁnite subgroup of R and let Φ be a normal free subgroup of R ∈ C = which is open, i.e., R/Φ . Then CΦ (H ) CΦ¯ (H ).

Proof Observe that H ∈ C since H is isomorphic to a subgroup of R/Φ.Wemay assume that R = HΦ = Φ H , because HΦ is open in R, so that its pro-C topology coincides with the topology induced from the pro-C topology of R (see Lemma 11.1.3(c)). We use induction on the order of H . Assume ﬁrst that H has prime order p; then, the result is the content of Lemma 13.1.3(b ). Assume next that H is cyclic and its order is not a prime number. Choose a maximal subgroup M of H . Then, since CΦ (H ) ≤ CΦ (M), one deduces that = ∩ = CΦ (H ) CΦ (M) CΦ (H ) CCΦ (M)(H ).

Note that CΦ (M) is open in Φ and so in R, and it is normal in R; hence the full pro-C topology of CΦ (M)H coincides with the topology induced by the topology of R (see Lemma 11.1.3(c)). Therefore, we may assume that Φ = CΦ (M) and

R = CΦ (M)H = ΦH = Φ H, so that M is a ﬁnite normal subgroup centralizing Φ. Thus, factoring M out and identifying Φ with its image modulo this factorization, it sufﬁces to prove the equal- = ity CΦ (H/M) CΦ (H/M). But this follows from the previous case. If H is noncyclic, take M1 and M2 to be two distinct maximal subgroups of H . By Proposition 13.1.4

CΦ (H ) = CΦ (M1) ∩ CΦ (M2) = CΦ (M1) ∩ CΦ (M2). ∩ Now, by the induction hypothesis, the latter expression coincides with CΦ¯ (M1) = CΦ¯ (M2) CΦ¯ (H ), as needed.

Corollary 13.1.6 Let R be a virtually free (or free-by-ﬁnite) abstract group. Let H1,H2 be ﬁnitely generated subgroups of R. Then

H1 ∩ H2 = H1 ∩ H2.

Theorem 13.1.7 Let R be a finitely generated free-by-C abstract group. Let H be a finitely generated subgroup of R which is closed in the pro-C topology of R. Then = ¯ NR(H ) NRCˆ (H). 376 13 Abstract Groups vs Their Profinite Completions

≤ ¯ Proof Obviously NR(H ) NRCˆ (H). We need to prove the reverse containment. = abs R = abs R By Proposition 8.2.3, we have that R Π1 ( ,) Π , where ( ,) is a graph of ﬁnite groups in C over a ﬁnite graph .LetSabs be the standard tree associated with this graph of groups and let S be the standard C-tree associated with (R,), considered as a graph of pro-C groups (see Sect. 6.5).

Case 1. H is infinite. By Proposition 8.2.4, Sabs has a unique minimal H -invariant subtree Dabs and its closure D = Dabs in S is the unique minimal H¯ -invariant C- subtree of S. ∈ ¯ = ¯ C If a NRCˆ (H), then aD D, because aD is also a minimal H -invariant - ¯ subtree of S. In other words, NRCˆ (H) acts on D; in particular, NRCˆ (H ) acts on D. abs Similarly, NR(H ) acts on D . Next we claim that the natural epimorphism of graphs \ abs −→ ¯ \ NR(H ) D NRCˆ (H) D is injective. Let s˜ = sΠabs(m), s˜ = s Π abs(m) ∈ Dabs (s, s ∈ R,m ∈ ), and as- ∈ ¯ ˜ = ˜ −1 abs = sume that there exists an a NRCˆ (H) such that s as. Then s asΠ (m) Π abs(m), i.e., s −1as ∈ Π abs(m) ≤ Π abs = R. Therefore, by Lemma 13.1.1, ∈ ∩ ¯ = a R NRCˆ (H) NR(H ), proving the claim. abs abs abs Since H \D is finite, so is NR(H )\D . Choose m ∈ , and t1 = 1Π (m), abs t2,...,tr ∈ D such that abs D = NR(H )t1 ∪. ···∪. NR(H )tr . Since this union is finite, taking closures we have = ∪···∪ = ¯ ∪···∪ ¯ D NR(H )t1 NR(H )tr NRCˆ (H)t1 NRCˆ (H)tr ≤ ¯ (the last equality holds since NR(H ) NRCˆ (H)). By the claim these unions are dis- = ¯ ∈ ¯ joint. So, in particular, NR(H )t1 NRCˆ (H)t1. Hence, if a NRCˆ (H), there exists a abs abs −1 abs b ∈ NR(H ) such that aΠ (m) = bΠ (m). Therefore b a = x ∈ Π (m) ≤ R. Using Lemma 13.1.1, ∈ ∩ ¯ = x R NRCˆ (H) NR(H ). ∈ ¯ ≤ Thus a NR(H ). This proves that NRCˆ (H) NR(H ), as required.

Case 2. H is finite. Observe that CRCˆ (H ) is the kernel of the natural homomorphism −→ NRCˆ (H ) Aut(H ); therefore, since Aut(H ) is finite, CRCˆ (H ) has finite index in

NRCˆ (H ), so that NRCˆ (H ) is finite if and only if CRCˆ (H ) is finite. Similarly, NR(H ) is finite if and only if CR(H ) is finite.

Subcase 2 (a). NR(H ) is ﬁnite. Let Φ be a normal free abstract subgroup of R such that R/Φ ∈ C. Then NΦ (H ) = Φ ∩ NR(H ) = 1; in particular CΦ (H ) = 1. By 13.1 Free-by-Finite Groups vs Their Proﬁnite Completions 377

= Corollary 13.1.5,CΦCˆ (H ) 1. Hence NRCˆ (H ) is ﬁnite. Then, by Theorem 7.1.2, = abs NRCˆ (H ) is conjugate to a subgroup of some vertex group Π(v) Π (v), so that we may assume that it is contained in Π(v). Thus = = = NRCˆ (H ) NΠ(v)(H ) NR(H ) NR(H ), proving the result in this case.

Subcase 2 (b). NR(H ) is infinite. Hence so is CΦ (H ). Since R is finitely generated, so is Φ. Note that CΦ (H ) is the subgroup of elements of the free group Φ fixed by the finite group H (as a group of automorphisms); therefore CΦ (H ) is a free factor of Φ (see Lemma 13.1.3(b)); hence CΦ (H ) is finitely generated; it is also closed in the pro-C topology of Φ, and so of R (see Lemma 11.1.2(d)). It follows that NR(H ) is finitely generated and closed in the pro-C topology of R. Therefore abs by Proposition 8.2.4 there is a unique minimal NΦ (H )-invariant subtree D of abs abs S whose closure D = D is the unique minimal NΦ (H )-invariant subtree of S; abs furthermore, NΦ (H )\D = NΦ (H )\D is finite. ¯ We claim that NRCˆ (H)also acts on D: since NΦ (H ) is infinite and the stabilizers of vertices of D are finite, we can apply Proposition 2.4.12 to conclude that D is the C = unique CΦ (H )-invariant pro- subtree of S. Now, by Corollary 13.1.5 CΦCˆ (H )

CΦ (H ), and again by Proposition 2.4.12, D is in fact the unique minimal NRCˆ (H )- invariant subtree of S. This proves the claim. Next we proceed as in Case 1: one sees as in that case that the natural map \ abs −→ \ = NR(H ) D NRCˆ (H ) D is injective; one deduces that NR(H )t1 NRCˆ (H )t1, = abs ∈ ≤ where t1 1Π (m),forsomem ; and this implies that NRCˆ (H ) NR(H ),as needed.

According to Theorem 11.2.2, finitely generated subgroups of a free-by-finite group are closed in its profinite topology. So the following result is a restatement of Theorem 13.1.7 when C is the class of all finite groups.

Corollary 13.1.8 Let R be a finitely generated free-by-finite (or virtually free) abstract group, and let H be a finitely generated subgroup. Then = ¯ NR(H ) NRˆ (H).

Proposition 13.1.9 Let R be an abstract residually C group. Let H = h be an ¯ = inﬁnite cyclic subgroup of R, and assume that NRCˆ (H) NR(H ). Then ¯ = (a) CRCˆ (H) CR(H ), and ∈ ¯ −1 = −1 = −1 (b) if γ NRCˆ (H), then either γ hγ h or γ hγ h .

Proof (a) Consider the natural homomorphism ∼ ϕ : NR(H ) −→ Aut(H ) = Z/2Z.

Then Ker(ϕ) = CR(H ). Note that ≤ ¯ ≤ ¯ = CR(H ) CRCˆ (H) NRCˆ (H) NR(H ). 378 13 Abstract Groups vs Their Proﬁnite Completions

Since the index of CR(H ) in NR(H ) is at most 2, the result follows immedi- ¯ = ∈ ∈ ¯ ately: suppose CRCˆ (H) NR(H ) and let r NR(H ); then r CRCˆ (H), and so ∈ = = ¯ r CR(H ); i.e., CR(H ) NR(H ); hence CR(H ) CRCˆ (H). (b) Consider the natural homomorphism : ¯ −→ ¯ ψ NRCˆ (H) Aut(H). = ¯ Then Ker(ψ) CRCˆ (H). Since CR(H ) has ﬁnite index in NR(H ), one obviously has [NR(H ) : CR(H )]≤[NR(H ) : CR(H )]. So, ¯ ¯ NR(H ) : CR(H ) = Im(ϕ) ≤ Im(ψ) = NR (H): CR (H) Cˆ Cˆ = NR(H ) : CR(H ) ≤ NR(H ) : CR(H ) . Therefore, Im(ψ) = Im(ϕ) ≤ Aut(H ). Since the elements of Aut(H ) are the identity and inversion, the result follows.

Putting together Theorem 13.1.7 and the proposition above we obtain

Corollary 13.1.10 Let R be an abstract ﬁnitely generated free-by-C group, and let H = h be an inﬁnite cyclic subgroup of R that is closed in its pro-C topology. Then ¯ = (a) CRCˆ (H) CR(H ), and ∈ ¯ −1 = −1 = −1 (b) if γ NRCˆ (H), then either γ hγ h or γ hγ h .

Lemma 13.1.11 Let R be a ﬁnitely generated abstract free-by-C group. Say Φ is a normal subgroup of R whichisfreeandR/Φ ∈ C. Let H be a cyclic subgroup of Φ, then

CR(H ) = CR Cl(H ) , where Cl(H ) denotes the closure of H in the pro-C topology of R.

Proof Note that Cl(H ) is also the closure of H in the pro-C topology of Φ, since Φ is closed in R. By Proposition 11.3.1 Cl(H ) is cyclic and contains H as a subgroup of ﬁnite index. Say Cl(H ) = x and H = xn.Now,ifa ∈ R and a−1xna = xn, then both a−1xa and x are n-th roots of xn. Since in a free abstract group n-th roots are unique, we deduce that a−1xa = x. Thus the result.

Corollary 13.1.12 Let R be an abstract ﬁnitely generated free-by-C group. Let H be a non-necessarily closed cyclic subgroup of Φ. Then ¯ = CRCˆ (H) CR(H ).

Proof By Lemma 13.1.11,CR(H )=CR(Cl(H )). Therefore, using Corollary 13.1.10 for the closed subgroup Cl(H ),

= = = ¯ CR(H ) CR Cl(H ) CRCˆ Cl(H ) CRCˆ (H), since H¯ = Cl(H ). 13.2 Polycyclic-by-Finite Groups vs Their Proﬁnite Completions 379

13.2 Polycyclic-by-Finite Groups vs Their Proﬁnite Completions

Lemma 13.2.1 Let R be a polycyclic-by-ﬁnite group. For each natural number n, n! put Rn = R . Then the normal subgroups Rt (t = 1, 2,...) form a fundamental system of neighbourhoods of the identity element 1 in the proﬁnite topology of R:

R = R1 ≥ R2 ≥···.

Proof First we prove by induction on the Hirsch number h(R) of R that Rm has ﬁnite index in R, for each natural number m = 1, 2,....Ifh(R) = 0, then R is ﬁnite and the assertion is clear. Assume h(R) ≥ 1; let A be a nontrivial free abelian normal subgroup of R (cf. Segal 1983, Chap. 1, Lemma 6). Then h(R/Am)

Lemma 13.2.2 Let R be a polycyclic-by-ﬁnite group and let H ≤ R. Then for each S f R, there exists some TS f R with TS ≤ S such that

NR(H TS/TS) ≤ SNR(H ).

Proof By Lemma 13.2.1 it sufﬁces to prove that for any given natural number s, there exists a natural number t(s) such that t(s) ≥ s and NR(H Rt(s)/Rt(s)) ≤ RsNR(H ). We shall prove this by contradiction. Suppose that for every natural number t ≥ s one has

NR(H Rt /Rt ) ≤ RsNR(H ).

Then for each t ≥ s, there exists some y(t) ∈ R − RsNR(H ) such that

y(t) ∈ NR(H Rt /Rt ).

Since R/Rs is ﬁnite, there exist inﬁnitely many t,sayt1

y(t1)Rs = y(t2)Rs =···; hence there exists

x ∈ R − RsNR(H ) (13.5) with

x = y(t1)z(t1) = y(t2)z(t2) =···, −1 where z(ti) ∈ Rs , for all i = 1, 2,.... Observe that x Hx ≤ HRs . Then the sub- −1 groups x Hx and H of HRs are conjugate in all ﬁnite quotients HRs/Rti of HRs : −1 = −1 indeed, x Hx z(ti) Hz(ti)(mod Rti ). By Theorem 11.4.5 they are conju- −1 = −1 ∈ −1 ∈ gate in Rs , i.e., x Hx rs Hrs ,forsomers Rs . Therefore, xrs NR(H ), and so x ∈ RsNR(H ), contradicting (13.5). 380 13 Abstract Groups vs Their Proﬁnite Completions

Lemma 13.2.3 Let G be a proﬁnite group and let H be a closed subgroup of G. Then NG(H ) = NG(H U/U). UoG

Proof Clearly NG(H ) ≤ NG(H U/U). UoG Conversely, let g ∈ N (H U/U). Then for every U G,wehave UoG G o g−1HgU = HU. Hence g−1Hg = g−1HgU = HU = H, U U i.e., g ∈ NG(H ).

Proposition 13.2.4 Let R be a polycyclic-by-ﬁnite group and let H ≤ R. Then ¯ = NRˆ (H) NR(H ).

¯ ≥ Proof Clearly NRˆ (H) NR(H ). To prove the reverse containment, first observe ˆ ¯ ˆ ¯ ˆ that for any T f R, one has R = RT , since R is dense in R and T is open in R. So, ¯ ¯ ¯ ¯ = ¯ ¯ ¯ = ¯ since T centralizes H T/T HT/T, we deduce that NRˆ (HT/T) NR(H T /T )T . Define N ={N | N f R}.ForN ∈ N ,letTN be as in Lemma 13.2.2. Then using the above observation and Lemma 13.2.3,wehave ¯ = ¯ ¯ = ¯ ¯ NRˆ (H) NRˆ (H)N NRˆ (H TN /TN ) N ∈N ∈N ∈N N N N ¯ ¯ = NR(H TN /TN )TN N = NR(H TN /TN )N ∈N ∈N N N ¯ ¯ ≤ NR(H )TN N = NR(H )N = NR(H ), N∈N N∈N where the first and last equalities are just the fact that the closure of a subgroup of a profinite group is the intersection of the open subgroups containing it.

From the above proposition and Proposition 13.1.9 one deduces the following corollary.

Corollary 13.2.5 Let R be a polycyclic-by-finite group and let H = h be an infinite cyclic subgroup of R. Then ¯ = (a) CRˆ (H) CR(H ), and ∈ ¯ −1 = −1 = −1 (b) if γ NRˆ (H), then either γ hγ h or γ hγ h . 13.2 Polycyclic-by-Finite Groups vs Their Profinite Completions 381

Remark 13.2.6 Part (a) of Corollary 13.2.5 can be improved to subgroups H that are not necessarily cyclic. In fact, one can prove that for any subgroups H and ¯ = K of R one has CK¯ (H) CK (H ) (cf. Ribes, Segal and Zalesskii 1998, Proposi- tion 3.3(a)).

Lemma 13.2.7 Let ϕ: R −→ S be a homomorphism of polycyclic-by-ﬁnite groups, and let ϕˆ: Rˆ −→ Sˆ denote the induced continuous homomorphism of the corresponding proﬁnite completions. Let H be a subgroup of S. Then ϕ−1(H ) =ˆϕ−1(H), where the closures ϕ−1(H ) and H are taken in Rˆ and Sˆ, respectively.

Proof We use induction on the Hirsch number h(S) of S.Ifh(S) = 0, then S is finite. Let K = Ker(ϕ) and let L be a finite subset of R such that ϕ(L) = H ∩ ϕ(R). Then K¯ = Kˆ = Ker(ϕ)ˆ (see Corollary 11.4.3), ϕ−1(H ) = LK and ϕˆ−1(H)¯ = LK¯ . So in this case the lemma holds. Suppose now that h(S) ≥ 1 and that the result holds whenever the Hirsch number of the codomain of the homomorphism is smaller that h(S).LetA be an infinite free abelian normal subgroup of S (cf. Segal 1983, Chap. 1, Lemma 6). Applying the induction hypothesis to the homomorphism ϕ R −→ S −→ S/A, we infer that ϕ−1(H A) = ϕ−1(HA). Since the closure of ϕ−1(H ) in Rˆ coincides − with the closure of ϕ 1(H ) in ϕ−1(H A) = ϕ−1(H A) and the closure of H in Sˆ coincides with the closure of H in HA= HA, (see Corollary 11.4.3(b)), we may replace R with ϕ−1(H A) and S with HA. Put N = H ∩A.IfN = 1, then h(S/N) < h(S); so, by induction applied to ϕ R −→ S −→ S/N, and the observation above, we get the desired result ϕ−1(H ) = ϕ−1(H). Finally, assume N = H ∩A = 1. Then S = AH , and, by Corollary 11.4.3, Sˆ = A¯ H¯ . From now on we use additive notation for the group A, but multiplicative notation for S.Ifx and y are elements of a group, we use the following notation xy = xyx−1. We think of A as a left S-module by means of the action s · a = sa (a ∈ A, s ∈ S). Similarly, A¯ is a profinite Sˆ-module by means of the (continuous) action s · a = sa (a ∈ A¯, s ∈ S¯). Consider the projection δ : S −→ A (if s = ah, with a ∈ A, h ∈ H , one puts δ(s) = a). Then δ extends uniquely to the projection δˆ : Sˆ −→ A.¯ ˆ It is well-known (and easy to check) that δ and δ are derivations: δ(s1s2) = δ(s1) + s ˆ ˆ s ˆ ˆ 1 δ(s2),fors1s2 ∈ S and δ(s1s2) = δ(s1) + 1 δ(s2),fors1s2 ∈ S. Consider the free abelian profinite group M = A ⊕ uZ of rank 1 + rank(A). Define a left multiplication of R on M as follows: if a ∈ A and r ∈ R, put r · a = 382 13 Abstract Groups vs Their Profinite Completions

(δϕ)(r)+u. One easily checks that this deﬁnes M as a left R-module. Put T = M R = (A ⊕ uZ) R. The proﬁnite completion of T is Tˆ = Mˆ Rˆ = (Aˆ ⊕ uZˆ ) Rˆ. Since M centralizes u, we have that the centralizer of u in T is MCR(u).Now C (u) = r ∈ R u = r · u = (δϕ)(r) + u = r ∈ R (δϕ)(r) = 0 R = r ∈ R ϕ(r) ∈ H = ϕ−1(H ). = −1 = ˆ −1 ¯ = ¯ −1 ¯ Therefore CT (u) Mϕ (H ). Similarly, CTˆ (u) Mϕ (H) Mϕ (H) (note that the topological closure M¯ of M in Tˆ coincides with Mˆ , by Corollary 11.4.3). By = ¯ −1 = ¯ −1 ¯ −1 ¯ ≥ Corollary 13.2.5(a), CT (u) CTˆ (u).SoMϕ (H ) Mϕ (H). Since ϕ (H) ϕ−1(H ) and ϕ−1(H)¯ ∩ M¯ = 1, we have ϕ−1(H)¯ = ϕ−1(H ), as needed.

We can now prove

Proposition 13.2.8 Let R be a polycyclic-by-ﬁnite group, and let H,K ≤ R. Then H ∩ K = H ∩ K, where the closures are taken in Rˆ.

Proof Let ι : H −→ R be the inclusion homomorphism. Then we have a commutative diagram H ι R

H = H ι R where, by Corollary 11.4.3, all the homomorphisms are inclusions. Observe that ι−1(K) = H ∩ K and ιˆ−1(K) = H ∩ K. Thus, the result follows from Lemma 13.2.7.

As a consequence we can reﬁne Proposition 13.2.4.

Corollary 13.2.9 Let R be a polycyclic-by-ﬁnite group and let H,K ≤ R. Then ¯ = NK¯ (H) NK (H ).

Proof Using Propositions 13.2.4 and 13.2.8 we have NK (H ) = NR(H ) ∩ K = ∩ ¯ = ¯ ∩ ¯ = ¯ NR(H ) K NRˆ (H) K NK¯ (H). Chapter 14 Conjugacy in Free Products and in Free-by-Finite Groups

In this chapter C denotes an extension-closed pseudovariety of ﬁnite groups

This chapter is concerned with conjugacy C-separability and subgroup conjugacy C-separability. It is shown that these properties are preserved by taking free products of abstract groups. We also show that free-by-C groups are both conjugacy C-separable and subgroup conjugacy C-separable. The basic tools for proving these results are related to the study of minimal invariant subtrees developed in Chap. 8.

14.1 Conjugacy Separability in Free-by-Finite Groups

First we show that conjugacy C-separability is preserved by taking free products.

Theorem 14.1.1 Let {Ri | i ∈ I} be a collection of abstract groups. Then the free product R = ∗ Ri is conjugacy C-separable if and only if each Ri is conjugacy i∈I C-separable.

Proof Assume that R is conjugacy C-separable. Since R is residually C,sois γ each Ri .Leta,b ∈ Ri and assume that a = b , where γ ∈ (Ri)Cˆ. Since Ri is a free factor of R,wehavethat(Ri)Cˆ ≤ RCˆ (see Lemma 11.1.3(b)). Hence γ ∈ RCˆ. c Therefore there exists some c ∈ R such that a = b . Finally, observe that c ∈ Ri (see, for example, Magnus, Karrass and Solitar 1966, Corollary 4.1.5). Thus Ri is conjugacy C-separable. Conversely, assume that each Ri is conjugacy C-separable. Then each Ri is residually C and hence so is R (cf. Gruenberg 1957, Theorem 4.1). Let a,b ∈ R with −1 a = γ bγ , where γ ∈ RCˆ. We need to prove that a and b are conjugate in R.Let J be a ﬁnite subset of I such that

a,b ∈ RJ = ∗ Ri ≤ ∗ Ri = R. i∈J i∈I

© Springer International Publishing AG 2017 383 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_14 384 14 Conjugacy in Free Products and in Free-by-Finite Groups ! = C Let γJ be the image of γ in (RJ )Cˆ j∈J (Rj )Cˆ (the free pro- product) under the projection

RCˆ → (RJ )Cˆ = (Rj )Cˆ; j∈J = −1 ≤ then a γJ bγJ . Since (RJ )Cˆ RCˆ, we may assume that I is ﬁnite. Consider ﬁrst the case when a is hyperbolic, i.e., a is not conjugate to any element in one of the factors Ri (i ∈ I ). Then according to Proposition 8.4.3(b) we have that a and b are conjugate in R, as needed. On the other hand, if a is not hyperbolic, then b is not hyperbolic either (Propo- −1 sition 8.4.3(a)). Say that a = xgix ,forsomex ∈ R, gi ∈ Ri and i ∈ I .Andsay −1 that b = yg˜j y , where y ∈ R, g˜j ∈ Rj and j ∈ I . Then −1 −1 −1 −1 yg˜j y = b = γaγ = γxgix γ , −1 −1 −1 so that g˜j ∈ Rj ∩ y γxRix γ y. It follows from Corollary 7.1.5 that i = j −1 and y γx ∈ (Ri)Cˆ. Therefore, since Ri is conjugacy C-separable, gi and g˜i are −1 −1 conjugate in Ri . Thus a = xgix and b = yg˜iy are conjugate in R.

Corollary 14.1.2 The free product of conjugacy separable abstract groups is conjugacy separable.

Since abstract abelian groups are obviously conjugacy C-separable, so is the free product of such groups. In particular, from Theorem 14.1.1 one deduces

Corollary 14.1.3 Abstract free groups are conjugacy C-separable.

Next we extend this corollary to free-by-C groups.

Theorem 14.1.4 Let R be a free-by-C abstract group. Then R is conjugacy C- separable.

Proof To ﬁx the notation, say that Φ R, where Φ is an abstract free group such ¯ that R/Φ ∈ C. Then we may think of ΦCˆ as the open subgroup Φ of RCˆ (see γ Lemma 11.1.3(b)). Let x,y ∈ R and let x = y, where γ ∈ RCˆ.Wehavetoshow ¯ that x and y are conjugate in R. We may assume that x = 1. Since RCˆ = RΦ,we have γ = rη,forsomer ∈ R, η ∈ Φ¯ . So replacing x with xr and γ with η,wemay ¯ assume that γ is in ΦCˆ = Φ. Then ¯ ¯ y ∈ xΦCˆ ∩ R = xΦ ∩ R = x(Φ ∩ R) = xΦ. ¯ Hence from now on we may also assume that R = xΦ. Note that RCˆ = xΦ. ¯ −1 −1 ¯ ¯ γ ¯ Since RCˆ/Φ is abelian, we have x γ xγ ∈ Φ, i.e., xΦ = x Φ. On the other hand ¯ we have that the natural map ρ : R/Φ −→ RCˆ/Φ is a bijection. Since ρ(yΦ) = 14.1 Conjugacy Separability in Free-by-Finite Groups 385 yΦ¯ = xγ Φ¯ = xΦ¯ = ρ(xΦ), we deduce that yΦ = xΦ. So from now on we assume that γ ¯ R = xΦ, y = x ∈ R, with γ ∈ Φ = ΦCˆ, and yΦ = xΦ. (14.1) Now we distinguish two cases.

Case 1. The order of x is inﬁnite. Let n be a positive integer such that xn ∈ Φ. So yn ∈ Φ and yn = (xn)γ . By Corollary 14.1.3 yn and xn are conjugate in Φ. Say f −1xnf = yn, where f ∈ Φ. Replacing x with fxf−1, we may assume that yn = xn. Therefore

∈ n = n ∩ γ CΦCˆ x CRCˆ x ΦCˆ.

Write Φ = Φ1 ∗ Φ2, where Φ1 is a free subgroup of Φ of ﬁnite rank such that n x ∈ Φ1. Then

ΦCˆ = (Φ1)Cˆ (Φ2)Cˆ = Φ1 Φ1, C n = n n = n the free pro- product. Note that CΦ (x ) CΦ1 (x ) and CΦCˆ (x ) C(Φ1)Cˆ (x ) (see Corollary 7.1.6(b)). Since Φ1 has ﬁnite rank we can use Corollary 13.1.12 to get that C (xn) = C (xn), and so Φ1 Φ1

n = n = n CΦ x CΦ x CΦCˆ x . n n n n n Therefore, γ ∈ CΦ (x ). Since CΦ (x ) ≤ CR(x ),wehaveCΦ (x ) ≤ CR(x ). n n Hence γ ∈ CR(x ). Thus x,y,γ ∈ CR(x ). n n n m n Since x = 1 and Φ is free, CΦ (x ) is cyclic, say CΦ (x ) = z and z = x , for some natural number m. Using the uniqueness of m-th roots in Φ, we get that n CR(x ) = CR(z). Hence x ∈ CR(z), i.e., x and z commute. n n Since R = xΦ, we obtain that CR(x ) = xCΦ (x ) = x z; therefore n n CR(x ) is abelian, and hence so is CR(x ). This implies that x = y; thus the result holds in this case.

Case 2. The order of x is ﬁnite. Observe that x is isomorphic to a subgroup of R/Φ, and so x∈C. We proceed by induction on the order of x. Subcase 2(a). The order of x is p, a prime. Then y is also of order p. By Theo- rem 10.9.3 the group R is a free product

R = xΦ = ∗(Ci × Φi) ∗ L, i∈I where L and each Φi are free groups and the Ci are groups of order p. Suppose x and y are not conjugate in R. Since every ﬁnite subgroup of R of order p is conjugate in R to one of the Ci (cf. Serre 1980, Part I, Corollary 1 of Proposition 2), = = ∈ = we may assume that Ci1 x and Ci2 y , where i1,i2 I and i1 i2. Hence = × ∗ × ∗ R (Ci1 Φi1 ) (Ci2 Φi2 ) R1, where

R1 = ∗ (Ci × Φi) ∗ L. i∈I−{i1,i2} 386 14 Conjugacy in Free Products and in Free-by-Finite Groups

˜ ˜ Define R = Ci ∗ Ci and let ϕ : R → R be a natural epimorphism that sends Ci 1 2 ˜ 1 and Ci2 identically to their corresponding copies in R and sends Φi1 , Φi2 and R1 C ˜ = to 1. Then x and y are not conjugate in the free pro- product RCˆ Ci1 Ci2 (see Corollary 7.1.6(b)). However, the epimorphism ϕ induces an epimorphism ϕˆ : → ˜ = ϕ(γ) = ˜ = RCˆ RCˆ Ci1 Ci2 , and so x y in RCˆ Ci1 Ci2 , a contradiction. Subcase 2(b). The order of x is finite but not a prime. Choose a natural number n such that the order of xn is a prime. By Subcase 2(a) above, replacing x by a certain conjugate in R, we may assume that xn = yn, and so γ centralizes xn; hence, by ∈ n = n = n Lemma 13.1.3(b ), γ CΦCˆ (x ) CΦ (x ). Put H x CΦ (x ). Since x normal- n n izes CΦ (x ), H is a subgroup of R. By Lemma 13.1.3(b), CΦ (x ) is a free factor n of Φ, and so it is closed in Φ. Hence CΦ (x ) is closed in R; moreover, the pro-C topology on it induced from R is its full pro-C topology (see Lemma 11.1.3(b)). ¯ Since x is finite, H is closed in R and HCˆ = H (see Corollary 11.2.5). Therefore,

¯ n HCˆ = H = xCΦ x .

It follows that x,y ∈ H and γ ∈ HCˆ. Hence we may assume that R = H = n n xCΦ (x ). Moreover, conditions (14.1) still hold, where now CΦ (x ) plays theroleofΦ. Note that then xn is a central subgroup of R, and R/ xn= n n n ( x/ x )CΦ (x ), where, with a certain abuse of notation, we identify CΦ (x ) with its isomorphic image in R/ xn. Denote by x˜ and y˜ the images of x and y in R/ xn, respectively. Then n n R/ x = ˜xCΦ x . Note that the order of x˜ is strictly smaller than the order of x; y˜ =˜xγ , with γ ∈ n n n ∼ CΦ (x ), and (R/ x )/CΦ (x ) = ˜x∈C. By the induction hypothesis, there exists n f f −1 some f ∈ CΦ (x ) such that y˜ =˜x . Replacing x with x and γ with f γ ,wemay n assume that y˜ =˜x; observe that conditions (14.1) still hold, with CΦ ( x ) playing n n n theroleofΦ. Therefore y = xc,forsomec ∈ x . Since xCΦ ( x ) = yCΦ ( x ), n and CΦ (x ) is a free group, we have c = 1. Thus x = y, and the result follows.

14.2 Conjugacy Subgroup Separability in Free-by-Finite Groups

Recall that an abstract group R is subgroup conjugacy C-separable if whenever H1 and H2 are ﬁnitely generated closed subgroups of R (in its pro-C topology), then H1 and H2 are conjugate in R if and only if their images in every ﬁnite quotient R/N ∈ C are conjugate.

Theorem 14.2.1 Let R be a ﬁnitely generated free-by-C abstract group. Then R is subgroup conjugacy C-separable.

Proof Let H1 and H2 be ﬁnitely generated closed subgroups of R. Since R is resid- −1 ually C, it sufﬁces to prove that if γ ∈ RCˆ and H1 = γ H2γ , then there exists some −1 r ∈ R such that H1 = rH2r . 14.2 Conjugacy Subgroup Separability in Free-by-Finite Groups 387

According to Proposition 8.2.3, R is the fundamental group of a graph of groups (R,) over a ﬁnite connected graph such that R(m) ∈ C for every m ∈ , and RCˆ is the pro-C fundamental group of (R,) viewed as a graph of pro-C groups. As usual, we denote by Sabs and S the abstract standard tree (or universal covering graph) and the standard C-tree, respectively, of (R,).

abs Case 1. H1 is infinite (hence so is H2). By Proposition 8.2.4, S has a unique min- abs = abs imal Hi -invariant subtree Di , and Di Di is the unique minimal Hi -invariant C-tree of S(i= 1, 2). Then γD2 is a minimal H1-invariant C-tree of S, and hence D1 = γD2. abs By Proposition 8.2.4(f), Di is an abstract connected component of Di consid- abs ered as an abstract graph, and any other component of Di has the form βDi ,for ∈ = abs some β Hi (i 1, 2). Therefore γD2 is an abstract connected component of D1. ˜ It follows that there exists some h1 ∈ H1 such that ˜ abs = abs h1γD2 D1 . ∈ abs abs Since H2 is infinite and the R-stabilizer of any m S is finite, the tree D2 ∈ abs ˜ ∈ abs ⊆ abs must contain at least one edge; say e E(D2 ). Then h1γe D1 S . Since abs ˜ R\S = RCˆ\S = , there exists some r1 ∈ R such that r1e = h1γe. Hence −1 ˜ r1 h1γ is in the RCˆ-stabilizer of e, which in fact coincides with the R-stabilizer −1 ˜ ∈ of e since this stabilizer is finite (see Sect. 6.5). Therefore r1 h1γ R, and so ˜ h1γ = r ∈ R. Finally, taking into account that H1 and H2 are closed, we have (see Lemma 11.1.1(c))

˜ −1 = ∩ = ∩ γ = ∩ (h1) r = ∩ r = r H2 R H2 R H1 R H1 R H1 H1 , as desired.

Case 2. H1 is ﬁnite (hence so is H2). Let Φ be a normal free subgroup of R such that ¯ ¯ R/Φ ∈ C. Since Φ is open in RCˆ, one has that RCˆ = RΦ. Hence γ = r γ , where ¯ γ r γ r r ∈ R and γ ∈ Φ. Then H2 = H = (H ) . So, replacing H1 with H and γ 1 ¯ 1 ¯ ¯ 1 with γ , we may assume that γ ∈ Φ. Hence ΦH1 = ΦH2, and so ΦH1 = ΦH2, because Φ is closed in R. Since in fact ΦHi is open in R,wehave(ΦHi)Cˆ ≤ RCˆ (i = 1, 2) (Lemma 11.1.3(c)). Thus from now on we may assume that

R = ΦH1 = ΦH2 = Φ H1 = Φ H2.

This implies that H1 and H2 are maximal ﬁnite subgroups of R, and so they are abs R-stabilizers of some vertices of S ⊆ S, say of v1 and v2, respectively (see, for example, Serre 1980, Theorem I.15; see also Theorem 7.1.2). As mentioned in Sect. 7.2, we may assume that the graph of groups (R,) is reduced; this means in this case that whenever e is an edge of which is not a loop, then the order of the ﬁnite group R(di(e)) is strictly smaller than the order of R(e) ¯ (i = 0, 1). Write R = RCˆ.Lete˜ be an edge of S and denote its image in by e. 388 14 Conjugacy in Free Products and in Free-by-Finite Groups

¯ ¯ ¯ Then, if v is one of the vertices of e˜ and their R-stabilizers are equal, Rv = Re˜,it must be because e is a loop of . Since H1 stabilizes γv2 and v1, it must stabilize every element of the chain ¯ [γv2,v1] in S (see Corollary 4.1.6); therefore, since H1 is maximal, it is the R- stabilizer of each element of the chain [γv2,v1]. In particular, the end points of any edge of this chain have the same R¯-stabilizers. By the comment above, this means that the projection on of any edge of [γv2,v1] must be a loop. Since the image of [γv2,v1] in is a connected subgraph of the ﬁnite graph , we deduce that the image of [γv2,v1] in has a unique vertex. Therefore γv2 and v1 areinthesame ¯ ¯ ¯ ¯ R-orbit. Hence Rv2 = Rγv2 = Rv1. abs ¯ Since R\S = R\S, one deduces that Rv2 = Rv1. Say rv2 = v1, where r ∈ R. = = −1 = −1 Then H1 Rv1 rRv2 r rH2r .

Since in a free-by-finite group every finitely generated subgroup is closed in the profinite topology (see Theorem 11.2.2), one has the following consequence.

Corollary 14.2.2 Let R be a finitely generated free-by-finite abstract group and let H1 and H2 be finitely generated subgroups of R. Then H1 and H2 are conjugate in ˆ R if and only if H1 and H2 are conjugate in R, or equivalently, if and only if the images of H1 and H2 are conjugate in every finite quotient of R. In this case this is equivalent to saying that R is subgroup conjugacy separable.

Next we show that subgroup conjugacy separability is preserved by taking free products.

Theorem 14.2.3 Let {Ri | i ∈ I} be a collection of abstract groups. Then

R = ∗ Ri i∈I is subgroup conjugacy C-separable if and only if each Ri is subgroup conjugacy C-separable.

Proof Assume first that R is subgroup conjugacy C-separable. Using an argument similar to the one used in the first part of the proof of Theorem 14.1.1 one shows that each Ri is subgroup conjugacy C-separable. Conversely, assume that each Ri is subgroup conjugacy C-separable. Let H1 and H2 be finitely generated subgroups of R that are closed in its pro-C topology. As- −1 sume that H1 = γ H2γ , where γ ∈ RCˆ. We need to show that then there exists −1 some r ∈ R such that H1 = rH2r . Since H1 and H2 are finitely generated, we may assume that the indexing set I is finite (see the argument in the proof of Theorem 14.1.1). Say I ={1,...,n} so that

R = R1 ∗···∗Rn.

If H1 is not contained in a conjugate of any of the free factors Ri , then the result is the content of Proposition 8.4.4 (observe that under this assumption one does not need to know that the free factors Ri are subgroup conjugacy C-separable). 14.3 Conjugacy Distinguishedness in Free-by-Finite Groups 389

−1 So assume that H1 ≤ a Rj a, where a ∈ R and j isaﬁxedindexinI . Conju- gating by a and renaming the free factors we may assume that H1 ≤ R1. Since

RCˆ = (R1)Cˆ ···(R1)Cˆ, = γ ≤ ∈ we deduce that H2 H1 (R1)Cˆ and γ (R1)Cˆ (see Corollary 7.1.5(a)). Since R1 is closed in the pro-C topology of R (see Lemma 11.1.2(d)), we have

H2 ≤ R ∩ (R1)Cˆ = R1. So the result in this case follows from the subgroup conjugacy C-separability of R1.

14.3 Conjugacy Distinguishedness in Free-by-Finite Groups

In this section we show that closed ﬁnitely generated subgroups of a free-by-C group are conjugacy C-distinguished.

Theorem 14.3.1 Let R be a free-by-C abstract group and let H be a ﬁnitely generated subgroup of R which is closed in its pro-C topology. Then H is conjugacy C-distinguished.

Proof This is equivalent to proving that if a ∈ R and aγ = γ −1aγ ∈ H¯ , where c −1 γ ∈ RCˆ, then there exist c ∈ R such that a = c ac ∈ H . It follows from a result of Scott (1974) that R is the abstract fundamental group Π abs = Π abs(R,)of a graph of groups (R,)over an abstract graph such that each vertex group R(v) is in C (v ∈ V()). Since R is free-by-C, there exists a subgroup Φ of R which is free and open in the pro-C topology of R. The pro-C topology of R induces on Φ its own full pro-C topology (see Lemma 11.1.3(b)).

Case (i). The element a has finite order. The pro-C topology of R induces on H its own full pro-C topology, so that one ¯ = can make the identification H HCˆ (see Corollary 11.2.5). Observe that H is also a free-by-C group. Since H is finitely generated, it is the fundamental group of a graph of groups in C, (R , ), over a finite graph (see Proposition 8.2.3). Hence ¯ H = HCˆ is the pro-C fundamental group of (R ,). In addition, we may make the identification R (v) = Π abs(v) (a subgroup of H ), for every vertex v of . Since −1 ¯ ¯ γ aγ ∈ H has finite order, it is conjugate in H = HCˆ to an element of some vertex abs group R (w) = Π (w) ≤ H (see Theorem 7.1.2). Therefore, since HCˆ ≤ RCˆ, a is conjugate in RCˆ to an element, say b,ofH . Thus, by Theorem 14.1.4, there exists a c ∈ R with c−1ac = b ∈ H .

Case (ii). The element a has inﬁnite order. 390 14 Conjugacy in Free Products and in Free-by-Finite Groups

By Theorem 11.2.4, Φ can be chosen so that Φ = (Φ ∩ H) ∗ L, where L is a ¯ ¯ subgroup of Φ. Note that RCˆ = RΦ since R is dense in RCˆ and Φ is open. So γ = ¯ −1 rγ1,forsomer ∈ R,γ1 ∈ Φ. Therefore, replacing a with r ar, we may assume ¯ that γ = γ1 ∈ Φ = ΦCˆ (for the last equality, see Lemma 11.1.3(c)). Since Φ has ﬁnite index in R,wehave1= an ∈ Φ, for some natural number n. Observe that ¯ ¯ Φ = ΦCˆ = (Φ ∩ H)Cˆ LCˆ = (Φ ∩ H) L, the free pro-C product. By Proposition 13.1.2(b), Φ ∩ H = Φ¯ ∩ H¯ ;soγ −1anγ ∈ Φ ∩ H . We deduce from Lemma 8.3.2 that an is nonhyperbolic as an element of the free product Φ = (Φ ∩ H) ∗ L; i.e., an is conjugate in Φ to an element of either Φ ∩ H or L; in fact it must be conjugate in Φ to an element of Φ ∩ H , since otherwise Φ ∩ H would contain a conjugate in Φ¯ of a nontrivial element of L¯ , which is not possible (see Corollary 7.1.5(a)). Say c−1anc ∈ Φ ∩H ,forsomec ∈ Φ. Then

γ −1c c−1anc c−1γ ∈ Φ ∩ H, and hence c−1γ ∈ Φ ∩ H (see Corollary 7.1.5(a)); we deduce that c−1γ ∈ H¯ . Since γ −1aγ ∈ H¯ ,wehave(γ −1c)c−1ac(c−1γ)∈ H¯ , and therefore c−1ac ∈ H¯ .Now, since H is closed in the pro-C topology of R by assumption, we have H¯ ∩ R = H . Thus c−1ac ∈ H¯ ∩ R = H, as needed.

In the profinite topology of a free-by-finite group every finitely generated subgroup is closed (see Theorem 11.2.2). Hence one has the following result.

Corollary 14.3.2 Let R be a free-by-ﬁnite abstract group, and let H be a ﬁnitely generated subgroup of R. Then H is conjugacy distinguished.

Remark 14.3.3 The condition in Theorem 14.3.1 that H is closed in the pro-C topology of R is necessary. For example, let R = Z, the free group of rank 1. Let p be a prime number and let C consist of all ﬁnite p-groups, so that the pro-C topology is, in this case, the pro-p topology. Consider the subgroup H = qZ of Z, where q is a prime, q = p. Then H is not closed in the pro-p topology of Z,butif n n ϕn : Z → Z/p Z is the natural epimorphism (n = 1, 2,...), then ϕn(H ) = Z/p Z. Therefore, if x ∈ Z−H , one has ϕn(x) ∈ ϕn(H ) for each n. Thus, since Z is abelian, H is not conjugacy C-distinguished in Z. Chapter 15 Conjugacy Separability in Amalgamated Products

In this chapter we study how conjugacy separability in abstract groups is preserved under the formation of certain free products with amalgamation. The main result (Theorem 15.9.2) shows that one can construct conjugacy separable groups by forming a free product amalgamating a cyclic subgroup of groups which are either finitely generated free-by-finite or polycyclic-by-finite; in fact one can iterate this process to obtain new conjugacy separable groups. In addition to conjugacy separability we consider in this chapter a whole array of other properties that are preserved by constructing amalgamated free products of groups with cyclic amalgamation, if one makes certain basic assumptions on the factors of the amalgamated free product. These properties are described in Sect. 15.3 and are satisfied, for example, by finitely generated free-by-finite groups and polycyclic-by-finite groups. The main tools in most results in this chapter are again related to the action of certain abstract groups on abstract trees and the action of certain profinite groups on profinite trees, and their inter-connections. In most cases in this chapter the pertinent groups are amalgamated free products and their profinite completions, and the pertinent trees and profinite trees are those canonically associated with amalgamated free products. As we know, an amalgamated free profinite product G = G1 H G2 is not proper in general, i.e., the canonical homomorphisms G1 → G and G2 → G are not necessarily injective (see Sect. 6.4). In this chapter the groups G1, G2 and H arise as profinite completions of abstract groups. For the general approach that is described in this chapter to work, one needs to make sure that the amalgamated free products of abstract groups that are of interest to us are first of all residually finite, and that, in addition, they satisfy sufficient properties to ensure that their standard trees are embedded in the standard profinite trees associated to the corresponding profinite completions (see Sect. 6.5). The crucial results that permit the use of these ideas are established in Sect. 15.1, particularly in Proposition 15.1.4.

© Springer International Publishing AG 2017 391 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0_15 392 15 Conjugacy Separability in Amalgamated Products

15.1 Abstract Free Products with Cyclic Amalgamation

Let G1 and G2 be proﬁnite groups with a common closed subgroup K. Then one can form the amalgamated free proﬁnite product

G = G1 K G2 of G1 and G2 amalgamating K, i.e., the pushout of the inclusion maps H → G1 and H → G2 in the category of proﬁnite groups (see Sect. 1.6). As pointed out in Sect. 6.4, the canonical homomorphisms G1 → G and G2 → G are not necessarily injective; if these canonical homomorphisms are injective, then we say that the amalgamated free proﬁnite product is proper (see Sect. 6.4), in which case the abstract amalgamated free product G1 ∗K G2 is naturally embedded in G = G1 K G2 (cf. RZ, Theorem 9.2.4). Then the following result is clear.

Lemma 15.1.1 Let G = G1 K G2 be a proper amalgamated free proﬁnite product of two proﬁnite groups G1 and G2 amalgamating a common closed subgroup K. Identify Gi with its canonical image in G(i= 1, 2). Then G1 ∩ G2 = K in G.

Next we study the proﬁnite topology of an amalgamated free product

R = R1 ∗H R2 (15.1) of two abstract groups R1 and R2 with a common subgroup H . We are interested in cases when R is residually finite (so that R ≤ Rˆ) and, in addition, the equality ˆ = R R1 Hˆ R2 (15.2) makes sense. The reason for seeking this is that for certain properties that we want to investigate in R, there are corresponding properties that are easier to handle in ˆ R, and hopefully translate back to R. When (15.2) holds and the subgroups R1, abs R2 and H are closed in R, the tree S of the amalgamated free product (15.1)is naturally embedded as a dense subgraph of the profinite tree S of the amalgamated free profinite product (15.2) (see Sect. 6.5). As we shall see in many of the results in this chapter, understanding the connections between R and Rˆ and between Sabs and S will be crucial. Recall that if R is a residually finite group and X ⊆ R, then X¯ denotes the topological closure of X in Rˆ.

Lemma 15.1.2 Let R1 and R2 be residually ﬁnite abstract groups with a common subgroup H , and let R = R1 ∗H R2 be their amalgamated free product amalgamating H . Assume that

(i) the profinite topology of R induces on each of R1, R2 and H its own full profi- ˆ ˆ ˆ ˆ nite topology, i.e., R1 ≤ R, R2 ≤ R and H ≤ Ri ⊆ R, and (ii) H is closed in the profinite topology of R1 and of R2. Then (a) R is residually finite; 15.1 Abstract Free Products with Cyclic Amalgamation 393

ˆ = = (b) R R1 Hˆ R2 R1 H¯ R2 (the amalgamated free proﬁnite product), and in particular the amalgamated free proﬁnite product R1 H¯ R2 is proper; (c) there are inclusions = ∗ ≤ ∗ ≤ = ˆ; R R1 H R2 R1 H¯ R2 R1 H¯ R2 R

(d) H , R1 and R2 are closed in the proﬁnite topology of R.

Proof (a) Let N be the collection of all normal subgroups of R of finite index. Put Ni ={Ri ∩ N | N ∈ N}; then, by assumption, Ni is a fundamental system of neighbourhoods of 1 in the profinite topology of Ri (i = 1, 2). SinceH is closed in the profinite topology of R , HN = H(i= 1, 2). Clearly N = 1 i N∈Ni N∈Ni (i = 1, 2) and N ∩R1 ∩H = N ∩R2 ∩H , for each N ∈ N . Therefore, R is residually finite (cf. Baumslag 1963, Corollary 2.41 or RZ, Exercise 9.2.7). (b) Note that, because of (a), the hypotheses mean that one can make the fol- ˆ ¯ ˆ lowing identifications: H = H , R1 = R1 and R2 = R2; and observe that R is topologically generated by R1 and R2. To prove (b) it only remains to show that the diagram R1

Hˆ Rˆ

R2 is a pushout diagram in the category of profinite groups; this amounts to proving the following (cf. Sect. 1.6): whenever A is a finite group, ψ1 : R1 −→ A and ψ2 : R2 −→ A are homomorphisms (necessarily continuous) such that ψ1(h) = ψ2(h) for all h ∈ H¯ , then there exists a (necessarily unique) homomorphism ψ : Rˆ −→ A extending ψ1 and ψ2. This easily follows from the fact that R = R1 ∗H R2: indeed, let ψ : R −→ A be the homomorphism of abstract groups that extends the restric- : ˆ −→ tions ψ1|R1 and ψ2|R2 .Letψ R A be the unique homomorphism extending ψ . Then clearly, ψ extends ψ1 and ψ2. (c) Since R1 H¯ R2 is proper, the natural homomorphism ∗ −→ = ˆ R1 H¯ R2 R1 H¯ R2 R is injective. Next consider the homomorphism : ∗ −→ ∗ η R1 H R2 R1 H¯ R2 induced by the inclusions Ri → Ri . We shall show that η is injective. To see this, let 1 = k ∈ R1 ∗H R2; then k can be written as a finite product

hw1w2w3 ···wn, (15.3) where h ∈ H , with h = 1ifn = 0, and where wj ∈ (R1 ∪R2)−H so that if wj ∈ R1 (respectively, if wj ∈ R2) then wj+1 ∈ R2 (respectively, wj+1 ∈ R1). The number 394 15 Conjugacy Separability in Amalgamated Products n is uniquely determined by k. Proving that η(k) = 1 requires us to show that the expression (15.3) satisfies analogous conditions if we now think of h as an element ¯ of H , and if wj ∈ Ri , then we think of wj as an element of Ri . In other words, we ¯ ¯ need to show that wi ∈/ H , for each i. This is indeed the case for if wi ∈ H , then we ¯ would have wi ∈ H ∩ (R1 ∪ R2) = H , since H is closed in the profinite topology of Ri (i = 1, 2), a contradiction. (d) We show first that H is closed in the profinite topology of R.Letk ∈ H¯ ∩ R. We must show that k ∈ H .Ifk/∈ H , then k, as an element of R = R1 ∗H R2, has a representation of the form k = w1w2 ···wn, with n ≥ 1, where wj ∈ (R1 ∪R2)−H , and if wj ∈ R1 (respectively, if wj ∈ R2) then wj+1 ∈ R2 (respectively, wj+1 ∈ R1). ¯ ¯ ¯ Note that wj ∈/ H(j= 1,...,n), since R1 ∩H = H = R2 ∩H , because, by hypothesis, H is closed in the profinite topologies of R1 and R2. Hence k = w1 ···wn is ∗ ≥ ¯ also a representation of k as an element of R1 H¯ R2. Since n 1, k cannot be in H , contradicting our assumption. Thus k ∈ H , as desired. Next we prove that R1 is closed; the case of R2 is analogous. Let k ∈ R1 ∩ R. ¯ We need to show that k ∈ R1. Since H is closed, we may assume that k/∈ H .Let k = w1w2 ···wn be a representation in the amalgamated free product R = R1 ∗H R2 ≥ ∗ as above with n 1. Again this is also a representation of k in R1 H¯ R2. Since k ∈ R1, the length of this representation must be 1. Thus, using Lemma 15.1.1 and the fact that H is closed, we have

k = w1 ∈ R1 ∩ (R1 ∪ R2) = (R1 ∩ R1) ∪ (R1 ∩ R2) = R1 ∪ (R1 ∩ R2 ∩ R2) ¯ = R1 ∪ (H ∩ R2) = R1 ∪ H = R1. This completes the proof of the proposition.

The following concepts play an important role in the study of the profinite topology on free products with cyclic amalgamation. Let R be an abstract group and let H be a cyclic subgroup of R. We say that R is H -potent if every subgroup of finite index of H is of the form H ∩ N for some normal subgroup N of finite index in R; R is H -quasi-potent if there exists a subgroup K of finite index in H such that R is K-potent. If R is H -potent for every cyclic subgroup H of R, one says that R is potent;ifR is H -quasi-potent for every cyclic subgroup H of R, one says that R is quasi-potent.

Lemma 15.1.3 (a) Polycyclic-by-ﬁnite groups are quasi-potent. (b) Free-by-ﬁnite groups are quasi-potent.

Proof The idea of the proof is essentially the same for both cases. We shall prove part (a) explicitly, and then we indicate the necessary changes to complete the proof for case (b). Let R be a polycyclic-by-finite group. Then R contains a normal subgroup P of finite index such that P is poly-(infinite cyclic), i.e., P admits a subnormal series all 15.1 Abstract Free Products with Cyclic Amalgamation 395 of whose intermediate quotients are infinite cyclic (cf. Segal 1983, Chap. 1, Propo- sition 2). Using the derived series for P , we deduce that R has a chain of normal subgroups

R ≥ A0 ≥ A1 ≥···≥Ak−1 ≥ Ak = 1 such that R/A0 is finite, and Ai/Ai+1 is free abelian (i = 0, 1,...,k − 1).Letx be an element of R of infinite order, and let s be the smallest positive integer with s s x ∈ A0. Say x ∈ Ai − Ai+1. Since Ai/Ai+1 is free abelian, there is some y ∈ Ai s t such that yAi+1 is part of a basis of Ai/Ai+1 and x Ai+1 = y Ai+1,forsome tn ={ tn | ∈ } natural number t.Nowletn be a natural number, and consider Ai a a Ai . = tn s Then Mn Ai Ai+1 is a normal subgroup of R; moreover, the order of x Mn in Ai/Mn is n. Therefore the order of xMn in R/Mn is sn, i.e., [ x: x∩Mn]= sn. Since R/Mn is polycyclic-by-finite, it is residually finite (see Theorem 11.4.1); 2 sn−1 hence there exists an N f R with Mn ≤ N such that x,x ,...x ∈/ N. Thus x∩N = xsn, proving part (a). In case (b), let R be an abstract group with a normal free subgroup Φ of finite index. In the proof above, we replace the groups Ai with the subgroups γi(Φ) of = the lower central series of Φ. Note that i γi(Φ) 1 and each γi(Φ)/γi+1(Φ) is free abelian (cf. Magnus, Karrass and Solitar 1966, Chap. 5, Corollary 5.12). Then the proof above goes through word by word in this case, with γi(Φ) playing the role of Ai ; the key point being that the quotient group R/Mn is residually finite since Mn is a verbal subgroup of Φ (see Example 11.2.7(d)).

Proposition 15.1.4 Let R1 and R2 be residually ﬁnite abstract groups with a common cyclic subgroup H = h, and consider their amalgamated free product

R = R1 ∗H R2 amalgamating H . Assume that

(i) R1 and R2 are quasi-potent, and (ii) H is closed in the proﬁnite topology of Ri (i = 1, 2). Then

(a) the profinite topology of R induces on R1, R2 and H their own full profinite topologies; (b) R is residually finite; ˆ = = (c) R R1 Hˆ R2 R1 H¯ R2; (d) there are inclusions = ∗ ≤ ∗ ≤ = ˆ R R1 H R2 R1 H¯ R2 R1 H¯ R2 R,

(f) H , R1 and R2 are closed in the proﬁnite topology of R.

Proof Let N denote the collection of all normal subgroups of R of finite index, and let Ni denote the collection of all normal subgroups of Ri of finite index (i = 1, 2). (a) Since Ri is assumed to be quasi-potent (i = 1, 2), its profinite topology induces on the cyclic group H its full profinite topology. So it suffices to show that 396 15 Conjugacy Separability in Amalgamated Products the profinite topology of R induces on R1 and R2 their own full profinite topologies. To see that this is the case we shall prove that if U1 f R1 and U2 f R2, then there exists some N ∈ N with N ∩ R1 ≤ U1 and N ∩ R2 ≤ U2. t t Say H ∩ U1 = h 1 and H ∩ U2 = h 2 , where t1 and t2 are natural numbers. Since Ri is quasi-potent, there exists ri such that for any given natural number ni , t r n the subgroup h i i i is of the form H ∩ Mi = H ∩ Ui ∩ Mi ,forsomeMi ∈ Ni (i = 1, 2). Choose n1 and n2 so that t1r1n1 = t2r2n2. Then t1r1n1 t2r2n2 H ∩ U1 ∩ M1 = h = h = H ∩ U2 ∩ M2. Consider the epimorphism : = ∗ −→ ˜ = ∩ ∗ ∩ ϕ R R1 H R2 R R1/(U1 M1) H/(H∩U1∩M1) R2/(U2 M2) induced by the canonical projections R1 −→ R1/(U1 ∩ M1) and R2 −→ R2/ ˜ (U2 ∩ M2). Since R1/(U1 ∩ M1) and R2/(U2 ∩ M2) are finite, the group R contains a free subgroup Φ of finite index (cf. Serre 1980, Chap. 2, Sect. 2.6, Proposition 12). In particular,

Φ ∩ R1/(U1 ∩ M1) = 1 = Φ ∩ R2/(U2 ∩ M2) . −1 Put N = ϕ (Φ). Then N ∈ N , and one has N ∩ R1 ≤ U1 and N ∩ R2 ≤ U2,as desired. In view of (a), parts (b), (c), (d) and (f) follow from Lemma 15.1.2.

15.2 Normalizers in Amalgamated Products of Groups

In this section we obtain a description of normalizers of certain cyclic (procyclic) subgroups of amalgamated free products of groups in terms of the normalizers in the factors.

Lemma 15.2.1

(a) Let G be a profinite group and let C1 and C2 be closed subgroups of a procyclic subgroup C of G. If C1 and C2 are conjugate in G, then C1 = C2. (b) Let R be a residually finite abstract group. Let C1 and C2 be subgroups of a cyclic subgroup C of R and assume that C1 and C2 are closed in the profinite topology of R. If C1 and C2 are conjugate in R, then C1 = C2. (c) Let R be a residually finite abstract group and let C1 and C2 be subgroups of a cyclic subgroup C of R. Assume that the profinite topology of R induces on C its full profinite topology. If C1 and C2 are conjugate in R, then C1 = C2.

Proof (a) Given any open normal subgroup U of G,wehavethatC1U/U and C2U/U are conjugate in G/U . Since they are subgroups of the ﬁnite cyclic group CU/U and they have the same order, we have C1U/U = C2U/U. Thus = = = C1 lim←− C1U/U lim←− C2U/U C2. U U 15.2 Normalizers in Amalgamated Products of Groups 397

ˆ (b) Note that the closures C1 and C2 in R of C1 and C2, respectively, are con- ˆ jugate in R.By(a),C1 = C2. Since C1 and C2 are closed in the proﬁnite topology of R, we deduce (see Lemma 11.1.1(c)) that

C1 = R ∩ C1 = R ∩ C2 = C2, as needed. (c) Let N be a normal subgroup of R of finite index. Then C1N/N and C2N/N are conjugate in R/N, and so they have the same order. Since they are subgroups of the finite cyclic group CN/N, it follows that C1N/N = C2N/N. One deduces that ˆ C1 = C2 in R. Therefore the closures of C1 and C2 in the profinite topology of R coincide:

Cl(C1) = R ∩ C1 = R ∩ C2 = Cl(C2). Since the profinite topology of R induces on C its full profinite topology, we have that the closures of C1 and C2 in the profinite topology of C coincide. However, since C is cyclic, every subgroup of C is closed in the profinite topology of C. Thus, C1 = C2.

Lemma 15.2.2 Let R be an abstract residually ﬁnite group that acts on an abstract tree T so that

(1) for every edge e of T , its stabilizer Re is a cyclic group, (2) the proﬁnite topology of R induces on each Re its full proﬁnite topology, and (3) = R\T is a tree.

Let H be a subgroup of Rv, for some vertex v of T . Then there exist graphs of groups (R,)and (R , ), where isasubtreeof, such that (a) R is the fundamental group of the graph of groups (R,)over : = abs R R Π1 ( ,), and (b) the normalizer NR(H ) of H in R is the fundamental group of the graph of groups (R , ):

= abs R NR(H ) Π1 , ,

where R (m) = NR(m)(H ), for all m ∈ .

H Proof First, we show that the normalizer NR(H ) acts on T , the subtree of ﬁxed H points of T under the action of H . Indeed, let g ∈ NR(H ) and let s ∈ T ; then if h ∈ H , one has hg = gh ,forsomeh ∈ H , so that h(gs) = gh s = gs, i.e., gs ∈ T H . To obtain the structure of NR(H ) we shall use the BassÐSerre theory (Serre 1980, H Theorem I.13). First we need to determine the quotient graph NR(H )\T . We claim that this quotient is a tree and it can be identiﬁed in a natural way with a subtree of R\T . Consider the natural morphism of graphs H ρ : NR(H )\T −→ = R\T. 398 15 Conjugacy Separability in Amalgamated Products

We shall show that ρ is injective. If T H does not have edges, then T H ={v};so H H NR(H )\T ={v}, which is obviously embedded in R\T . Suppose then that T H contains edges. Since NR(H )\T is connected and R\T is a tree, to show that ρ H is injective it suffices to prove that it is injective on the set of edges of NR(H )\T . Assume that e is an edge of T H such that ge ∈ T H ,forsomeg ∈ R. We need to show the existence of an x ∈ NR(H ) with xe = ge; in fact, we shall show that x −1 can be taken to be g. First observe that the R-stabilizer of ge is precisely gReg . −1 −1 Hence H and gHg are subgroups of gReg . Since the profinite topology of R −1 −1 induces on gReg its full profinite topology by hypothesis, we have H = gHg , according to Lemma 15.2.1.Sog ∈ NR(H ), as asserted. This proves the claim. Hence one has a commutative diagram T = R\T

H H T = NR(H )\T where the horizontal maps are the canonical projections. H Put = NR(H )\T . Then is a subtree of by the claim above. Let Σ be a connected transversal of in T H . Extend Σ to a connected transversal Σ = abs R of in T . So according to Theorem I.13 in Serre (1980), R Π1 ( ,) and N (H ) = Π abs(R , ), where R 1 R = = ∩ = = (m) NR(H ) s NR(H ) Rs NRs (H ) NR(m)(H ), with m ∈ , s ∈ Σ and Rs = m (here we identify R(m) with its canonical image = abs R Rs in R Π1 ( ,)). The following lemma is an analogue of Lemma 15.2.2 in the context of proﬁ- nite groups. The proof uses the same line of argument, but instead of appealing to Theorem I.13 in Serre (1980), one uses Theorem 6.6.1.

Lemma 15.2.3 Let G be a proﬁnite group that acts continuously on a proﬁnite tree T so that

(1) for every edge e of T , its stabilizer Ge is a procyclic group, and (2) = G\T is a ﬁnite tree.

Let H be a closed subgroup of Gv, for some vertex v of T . Then there exist graphs of profinite groups (G,) and (G , ), where is a subtree of , such that (a) G is the profinite fundamental group of the graph of groups (G,)over : G = Π1(G,), and (b) the normalizer NG(H ) of H in G is the profinite fundamental group of the G graph of groups ( ,):

NG(H ) = Π1 G , ,

where G (m) = NG(m)(H ), for all m ∈ . 15.3 Conjugacy Separability of Amalgamated Products 399

Proposition 15.2.4

(a) Let R = R1 ∗H R2 be an amalgamated free product of abstract groups R1 and R2 with amalgamated cyclic subgroup H , and assume that R is residually finite. Furthermore, assume that the profinite topology of R induces on H its full profinite topology. Let h ∈ H . Then

= ∗ NR h NR1 h H NR2 h .

(b) Let G = G1 H G2 be a proper amalgamated free profinite product of profinite groups G1 and G2 with amalgamated procyclic subgroup H . Let h ∈ H . Then = NG( h ) NG1 ( h ) H NG2 ( h ), and this amalgamated free profinite product is proper.

Proof (a) Consider the standard tree T associated with the amalgamated free product R = R1 ∗H R2 such that R acts on T with fundamental domain a segment T of T (corresponding to R\T ):

e T = v1 v2 with vertices v1,v2 and edge e in such a way that the stabilizers of its vertices are = = = Rv1 R1 and Rv2 R2, and the stabilizer of its edge is Re H (cf. Serre 1980, Theorem I.7). Since h ﬁxes the segment T , the result is clear from the proof of h Lemma 15.2.2: in this case NR( h)\T = T , and

= ∗ = ∗ NR h NR1 h NH ( h) NR2 h NR1 h H NR2 h , since H is abelian. (b) The proof in this case is similar: one considers the standard proﬁnite tree associated with the proper amalgamated free proﬁnite product G = G1 H G2 (see Example 6.2.3(d) and Corollary 6.3.6). Now, since

G = G1 H G2 ≤ ≤ is proper, NGi ( h ) NG( h ), and since H is abelian, H NGi ( h ); it follows = that NG( h ) NG1 ( h ) H NG2 ( h ). One then has that this amalgamated free proﬁnite product is proper (cf. RZ, Theorem 9.2.4).

15.3 Conjugacy Separability of Amalgamated Products

The purpose of this and the next sections is to show how one can construct further examples of abstract groups that are conjugacy separable using certain free constructions. In Corollary 14.1.2 we saw that conjugacy separability is preserved by taking free products. Here we explore whether conjugacy separability is preserved by taking amalgamated free products. An obvious difficulty is that in general an amalgamated free product of residually finite groups need not be residually finite. 400 15 Conjugacy Separability in Amalgamated Products

We show that the amalgamated free product of groups that are either finitely generated free-by-finite or polycyclic-by-finite amalgamating a common cyclic subgroup is conjugacy separable; furthermore, we can iterate this construction and obtain again conjugacy separable groups (see Theorem 15.9.2). To control this iterative process we isolate some properties that we shall show are preserved by free products with cyclic amalgamation. Next we describe these properties, and we denote by X the class of abstract groups that satisfy them.

15.3.1 The Class X . By X we denote a class of abstract groups that is defined as follows. An abstract group R is in X if (a) R is conjugacy separable; (b) R is quasi-potent; (c) whenever A and B are cyclic subgroups of R,thesetAB is closed in the profinite topology of R; (d) every cyclic subgroup of R is conjugacy distinguished, i.e., if C is a cyclic ˆ subgroup of R and a ∈ R, then aR ∩ C =∅if and only if aR ∩ C¯ =∅; (e) if A and B are cyclic subgroups of R, then A ∩ B = 1 if and only if A¯ ∩ B¯ = 1; and = ∈ ˆ ∈ ¯ (f) if A a is an infinite cyclic subgroup of R, and γ R with γ NRˆ (A), then ∈ −1 ∈{ −1} γ NRˆ (A), i.e., γaγ a,a . We remark that the groups in X are residually finite (because of property (a)) and in particular R ≤ Rˆ; this guarantees that the statements of properties (d), (e) and (f) make sense. If R ∈ X , then according to property (b) the profinite topology of R induces the full profinite topology on any cyclic subgroup C of R; i.e., C¯ = Cˆ . It follows from property (c) that every cyclic subgroup of R ∈ X is closed in the profinite topology of R. Finally, we observe that for groups in X , property (e) is equivalent to A ∩ B = A¯ ∩ B¯ (see Lemma 15.7.1).

Example 15.3.2 (a) Finitely generated free-by-finite abstract groups are in X . To verify this see Theorem 11.3.8, Corollary 13.1.6, Corollary 13.1.10, Theorem 14.1.4, Corol- lary 14.3.2 and Lemma 15.1.3. In fact, among these results only Corol- lary 13.1.10 (corresponding to property (f)) requires finite generation, and most likely this restriction is unnecessary. (b) Free groups of any rank are in X . From the comments in part (a), it suffices to show that to verify property (f) of X in the case of a free group Φ of arbitrary rank, one may assume in fact that Φ has finite rank. Indeed, let 1 = a ∈ Φ. Then there exists a decomposition Φ = Φ1 ∗ Φ2 such that a ∈ Φ1 and rank(Φ1)<∞. ˆ ˆ −1 Note that Φ = Φ1 Φ2 = Φ1 Φ2. So, if γ ∈ R and γ aγ = a, then one deduces from Corollary 7.1.5(a) that γ ∈ Φ1. But then property (f) follows from Corollary 13.1.10. (c) Polycyclic-by-finite groups are in X . To verify this see Theorem 11.4.4, Propo- sition 11.4.6, Proposition 11.4.7, Corollary 13.2.5 and Proposition 13.2.8. (d) Free abelian groups of any rank are also in the class X , as one easily checks. 15.3 Conjugacy Separability of Amalgamated Products 401

The importance of the class X is borne out by the following result that we prove in Theorem 15.9.1: the class X is closed under taking free products with cyclic amalgamations. The proof of this requires a series of results to show that each of the properties (a)Ð(f) that define the class X is preserved by taking amalgamated free products of groups in the class with cyclic amalgamation; this is done in this and the following sections. The proofs of these results illustrate the general method that lies at the heart of most applications to abstract groups in this book, namely (I) the interaction between the abstract fundamental groups Π abs of a graph of groups (G,)of a certain type and its profinite completion, and (II) the interaction between the universal covering graph Sabs (a tree) of a certain graph of abstract groups (G,) over a finite connected graph and the profinite universal covering graph (a profinite tree) of the corresponding graph of profinite groups (G¯,), where G¯(m) = G(m), for each m ∈ .

15.3.3 General Remarks. Here we point out a series of facts that will be used freely without further reference in the remainder of this chapter. The general set-up is the following: the graph of abstract groups (G,)of interest is R1• H •R2

It is assumed that R1 and R2 are residually finite and quasi-potent; we also assume that H is a common cyclic subgroup of R1 and R2 which, moreover, is closed in the profinite topologies of both R1 and R2 (observe that these properties are satisfied by the groups in X ). This means that the hypotheses of Proposition 15.1.4 hold and hence its conclusions. Therefore the corresponding graph (G¯,)of profinite groups is R1• Hˆ •R2

Furthermore, = ∗ = abs G R R1 H R2 Π1 ( ,) ˆ = = = G¯ ∩ = is residually finite, R R1 Hˆ R2 R1 H¯ R2 Π1( ,), and Ri R Ri (i = 1, 2). It follows (see Propositions 6.5.3 and 6.5.4) that the tree Sabs = Sabs(R) associated with the amalgamated free product decomposition R = R1 ∗H R2 is a dense subgraph of the profinite tree S = S(R)ˆ associated with the amalgamated free ˆ = profinite product decomposition R R1 H¯ R2. abs In the graphs S ⊆ S, we shall denote by v1 and v2 the vertices 1R1 = 1R1 abs and 1R2 = 1R2, respectively, and e ∈ S ⊆ S will denote the edge with vertices v1 and v2. abs Recall that the vertices of S have the form v = rvi (r ∈ R) (i = 1, 2), and the edges of Sabs have the form e = re (r ∈ R).TheR-stabilizers of these vertices = −1 = = −1 and edges are Rrvi rRir (i 1, 2) and Rre rHr . Therefore, if in addition R1 and R2 satisfy any one of the properties (a)Ð(f) in 15.3.1, then so does each of abs the R-stabilizers of the vertices S . In particular, if R1 and R2 are in the class X , 402 15 Conjugacy Separability in Amalgamated Products

abs so is each R-stabilizer Rv for v ∈ V(S ). Moreover, Rv and Re areclosedinthe proﬁnite topology of R for each v ∈ V(Sabs) and e ∈ E(Sabs). Similarly the corresponding Rˆ-stabilizers of the vertices and edges of Sabs are ˆ = −1 ∈ = ˆ = ˆ −1 ∈ (R)rvi rRir (r R) (i 1, 2) and (R)re rHr (r R). Therefore, using ˆ , ˆ Proposition 15.1.4, we deduce that (R)rv = Rrv = Rrv (i = 1, 2) and (R)re = , i i i Rre = Rre (r ∈ R).

We begin with the following result, which motivates in part the introduction of the class X .

Theorem 15.3.4 Let R1 and R2 be groups in X and let R = R1 ∗H R2 be their amalgamated free product with amalgamated common cyclic subgroup H . Then R is conjugacy separable.

Proof Let a,b ∈ R, γ ∈ Rˆ and assume that γ −1bγ = a; then we must show that there exists some r ∈ R such that r−1br = a. According to Proposition 15.1.4, = ∗ ≤ ∗ ≤ = ˆ R R1 H R2 R1 H¯ R2 R1 H¯ R2 R, ¯ ˆ where R1, R2 and H are the closures in R of R1, R2 and H , respectively. We consider the abstract tree Sabs = Sabs(R) and the proﬁnite tree S = S(R)ˆ associated with the given amalgamated free product decompositions of R and Rˆ (see 15.3.3).

Case 1. The element a is not hyperbolic (with respect to the action of R on the tree abs S (R)). In other words, a is in a conjugate in R of either R1 or R2. By Proposi- tion 8.3.5(a), b is not hyperbolic either. Replacing a and b with appropriate conjugates in R, we may assume that a,b ∈ R1 ∪ R2. Say a ∈ Ri and b ∈ Rj (i, j ∈{1, 2}).Ifγ ∈ Ri , we also have that b ∈ Ri ; then, because Ri is conjugacy separable, we deduce that a and b are conjugate in Ri , and a fortiori in R, as needed. Similarly, if γ ∈ Rj , then a and b are conjugate in R. So from now on we shall assume that γ/∈ Ri ∪Rj . Then using Corollary 7.1.5(b), ∩ −1 ≤ ¯ −1 ∈ Ri γ Rj γ γ1Hγ1 , where γ1 Ri, and ∩ −1 ≤ ¯ −1 ∈ Rj γ Riγ γ2Hγ2 , where γ2 Rj . ∈ ¯ −1 ∈ ¯ −1 ∈ X Hence, a γ1Hγ1 and b γ2Hγ2 . Since R1,R2 and H is cyclic, it follows from property (d) for the groups in X that a is conjugate in Ri to an element of H , and b is conjugate in Rj to an element of H . Hence we may assume that ∈ = ∈ a,b H . So, by Lemma 15.2.1(a), a b . We deduce that γ NRˆ ( a ).Now, by Proposition 15.2.4(b),

N ˆ a = N a ¯ N a . R R1 H R2 For i = 1, 2, consider the natural continuous homomorphism

ϕ : N a −→ Aut a i Ri 15.3 Conjugacy Separability of Amalgamated Products 403 that sends α ∈ N ( a) to the automorphism of a defined by x → αxα−1 Ri (x ∈ a). Observe that the homomorphisms ϕ1 and ϕ2 are trivial when restricted ¯ to H , since this group is abelian and a ∈ H . Note that the image of ϕi (i = 1, 2) is in Aut( a), and so it is finite: indeed, if a has finite order, this is clear, and if a has infinite order, the statement follows from property (f) of the groups in X . Therefore, by the universal property of amalgamated free profinite products, ϕ1 and ϕ2 extend : −→ uniquely to a continuous homomorphism ϕ NRˆ ( a ) Aut( a ). Since the amalgamated free product N ( a) ¯ N ( a) is proper, we deduce R1 H R2 that the abstract amalgamated free product N ( a) ∗ ¯ N ( a) is residually finite R1 H R2 and it is embedded densely in N ( a) ¯ N ( a) = N ˆ ( a). Since Aut( a) is R1 H R2 R − finite, there exists some c ∈ N ( a)∗ ¯ N ( a) with ϕ(c) = ϕ(γ). Then cac 1 = R1 H R2 γaγ−1 = b. Write

c = c1c2 ···ct −1 with c ∈ N ( a) ∪ N ( a) (i = 1,...,t). Define a = c a + c (i = 1,...,t), i R1 R2 i i i 1 i where at+1 = a. Observe that ai ∈ a≤H : indeed, if H is finite, then each ci −1 normalizes a; and if H is infinite, then ai ∈{a,a }, by assumption (f) on the class X . Since R1 and R2 are conjugacy separable, there exist elements r1,...,rt ∈ R1 ∪ R2 such that = −1 = ai riai+1ri (i 1,...,t). −1 Define r = r1 ···rt ; then rar = b, and r ∈ R, as needed.

Case 2. The element a is hyperbolic. The idea of the proof in this case is to show that there exists a certain polycyclic subgroup P of R such that a,b ∈ P and γ ∈ Pˆ, so that we can take advantage of the conjugacy separability of polycyclic groups (see Theorem 11.4.4). According to Proposition 8.3.5, b is also hyperbolic. Consider the Tits lines La abs and Lb in S corresponding to a and b, respectively. By Proposition 8.1.3(c), La = aT1 and Lb = bT2, where T1 =[w1,aw1] and T2 =[w2,bw2], for any ¯ abs abs w1 ∈ La and w2 ∈ Lb.Lete = 1H = 1H ∈ S ⊆ S, the edge of S (respectively, S) stabilized by H (respectively, H¯ ). We assert first that one may assume abs that e ∈ T1. By the definition of S there exists some r ∈ R such that e ∈ rT1. Define a = rar−1; then b = γr−1a (γ r−1)−1. Note that a is also hyperbolic with

= = corresponding Tits line La rTa (see Lemma 8.1.6); clearly La a T1, where = ∈ −1 T1 rT1 and e T1. Hence, replacing a with a and γ with γr , the assertion follows. So from now on we assume that e ∈ T1. Furthermore, by Proposition 8.3.5(b), ¯ ˆ we may assume that γ ∈ H = (R)e = Re.

Next we claim that, in fact, whenever [w,w ] is a ﬁnite chain in La (w, w ∈

V(La)) such that e ∈[w,w ], then we may assume that γ ∈ R[w,w ], where R[w,w ] is the subgroup of R that stabilizes all the elements of [w,w ]. We show this by induction on the length of such [w,w ]; from above, this property holds when the length of [w,w ] is 1, i.e., when [w,w ] contains only one edge, namely e.Soto prove this claim it sufﬁces to show that if the property holds for any such a chain 404 15 Conjugacy Separability in Amalgamated Products

E in La (i.e., e ∈ E and γ ∈ RE ), and if e1 is an edge in La − E that has a vertex, say v,inE, then for the chain ˜ E = E ∪ d0(e1), d1(e1), e , ∈ = one can also assume that γ RE˜ . To verify this, deﬁne e2 γe1. By Proposi- tion 8.3.5(a), e2 ∈ γ La = Lb. We assert that e2 ∈ Lb. Indeed, observe that γv= v, since γ ∈ RE by assumption; so e1 and e2 have v as a common vertex; therefore abs v ∈ Lb ∩ S = Lb (see Proposition 8.3.1(b)); so e2 belongs to the abstract connected component of Lb in Lb, which is Lb (see Proposition 8.3.1(e)); thus e2 ∈ Lb, verifying the assertion. Next, from the deﬁnition of Sabs, we know that there exists some r ∈ R such that re1 = e2; since v is a vertex of both e1 and e2, we deduce that −1 −1 r ∈ Rv,theR-stabilizer of v. Since γ e2 = e1 = r e2, we have that = −1 ∈ ˆ = γ1 γr (R)e2 Re2

(see 15.3.3). As pointed out in 15.3.3, Rv is in X and it is closed in the profinite topology of R. Now, since Re and RE are cyclic subgroups of Rv,wehave ∩ = Re2 RE Rv Re2 RE, X = −1 = −1 according to property (c) of the groups in . Hence r γ1 γ c1c2 , where ∈ ∈ ˜ = c1 Re2 and c2 RE. Define γ c2γ . Then −1 −1 γe˜ 1 = c2γe1 = c2e2 = r c1e2 = r e2 = e1. ˜ ∈ ˆ = ˜ ∈ So γ (R)e1 Re1 (see 15.3.3). Since clearly we also have that γ RE, we deduce ˜ ∈ −1 ˜ that γ RE˜ . Thus, replacing b with c2 bc2 and γ with γ , the claim is proved. Consider now a finite chain E in La containing e and ae. Define P = RE ∩ −1 aREa . Then P is a cyclic subgroup of H and clearly −1 a Pa≤ RE ≤ Re = H. −1 Since R1,R2 ∈ X , it follows from Proposition 15.1.4 that P and a Pa are closed in the profinite topology of R; then from Lemma 15.2.1 one has that P = a−1Pa. Hence a normalizes P . Define P1 = P,a. As shown above we may assume that ¯ ¯ γ ∈ P ≤ P1. Observe that P1 is polycyclic. By property (b) of the groups in X , P¯ = Pˆ. Since a is hyperbolic, a= a (see Lemma 8.3.2). Next we claim that P1 = P1. To see this assume first that P ∩ a=1; then P1 = P a; therefore, using Corollary 11.4.3(b), ˆ ¯ P1 = P a=P a=P1.

If, on the other hand, P ∩ a=1, then P has finite index in P1, and since the profinite topology of R induces on P its full profinite topology, it also induces on P1 its full profinite topology, so that P1 = P1, proving the claim. ¯ Hence γ ∈ P1. Now note that P1/P is abelian, and so −1 −1 −1 ¯ ab = aγa γ ∈ P ∩ R = P ≤ P1, since P is closed. Therefore we also have b ∈ P1. Finally, since P1 is polycyclic, it is conjugacy separable according to Theorem 11.4.4, and thus we have that a and b are conjugate in P1, and so in R, as desired. 15.4 Amalgamated Products, Quasi-potency and Subgroup Separability 405

As we have pointed out above, polycyclic-by-finite and finitely generated free- by-finite groups belong to the class X . Hence we have the following result.

Corollary 15.3.5 Let R = R1 ∗H R2 be an amalgamated free product of two abstract groups R1 and R2 with amalgamated common cyclic subgroup H . Assume that R1 and R2 are either polycyclic-by-finite or finitely generated free-by-finite. Then R is conjugacy separable.

15.4 Amalgamated Products, Quasi-potency and Subgroup Separability

Here we show that quasi-potency (property (b) of the class X ) is preserved by taking amalgamated free products with cyclic amalgamation (under some mild conditions). In general subgroup separability in groups (the closedness of ﬁnitely generated subgroups) is preserved when taking free products (Romanovskii 1969; Burns 1971), but not amalgamated free products (Rips 1990). However, if one is only interested in ‘cyclic subgroup separability’, then we show here that this property is preserved by amalgamated free products with cyclic amalgamations (a group is cyclic subgroup separable if its cyclic subgroups are closed in its proﬁnite topology) if one assumes in addition that the factors are quasi-potent.

Proposition 15.4.1 Let R = R1 ∗H R2, where H is a common cyclic subgroup of the abstract groups R1 and R2. Assume that R1 and R2 are residually ﬁnite and quasi-potent; furthermore, assume that H is closed in the proﬁnite topology of both R1 and R2. Then (a) R is quasi-potent; (b) if, in addition, R1 and R2 are cyclic subgroup separable, then R is also cyclic subgroup separable.

Proof By Proposition 15.1.4, R = R1 ∗H R2 is a residually finite group; moreover, ˆ = = R R1 Hˆ R2 R1 H¯ R2 and in particular this amalgamated free profinite product is proper. As usual, we denote by Sabs = Sabs(R) the standard abstract tree associated with the above amalgamated free product decomposition of R; and by S = S(R)ˆ the standard profinite tree associated with the decomposition of Rˆ as an amalgamated free profinite product as above (see 15.3.3). Let x ∈ R be an element of infinite order.

Case 1. The element x is not hyperbolic (with respect to the action of R on S(R)), −1 i.e., x is in a conjugate of either R1 or R2. Say x ∈ gR1g ,forsomeg ∈ R. Since = −1 ∗ −1 R gR1g gHg−1 gR2g , we may assume that x ∈ R1. By Proposition 15.1.4, the profinite topology of R induces on R1 its full profinite topology and R1 is closed; therefore, if R1 is cyclic 406 15 Conjugacy Separability in Amalgamated Products subgroup separable, x is closed in the profinite topology of R1, and so in the profinite topology of R. This proves part (b) in this case. t t Let H = h. Then there exist natural numbers t1 and t2 such that h 1 and h 2 are potent subgroups of R1 and R2, respectively. Let s be a common multiple of t1 s s and t2. Choose M1 f R1 and M2 f R2 so that M1 ∩ H = h and M2 ∩ H = h ; consider the natural epimorphism = ∗ −→ %= ∗ ϕ: R R1 H R2 R R1/M1 HM1/M1 R2/M2. Note that R˜ is residually finite (cf. Baumslag 1963, Theorem 3, or Serre 1980, % % Proposition II.11). Let M f R be a normal subgroup of R of finite index with −1 trivial intersection with R1/M1 and R2/M2; put N = ϕ (M), then clearly N f R and, since (H M /M ) ∩ M = 1, 1 1 s N ∩ H = a ∈ H ϕ(a) ∈ M = a ∈ H ϕ(a) = 1 = M1 ∩ H = h . Let e be the order of x in R/N. Choose a natural number m such that xm is k potent in R1, and set k = me. We claim that x is potent in R.Forlett be a tk tk natural number; choose L1 f R1 so that L1 ∩ x= x ; since x ∈ N,wehave tk L1 ∩ N ∩ x= x . To complete the verification of the claim we must show that tk there exists some S f R with S ∩ x= x ; and for this, it suffices to show that there exists such an S with S ∩ R1 = L1 ∩ N. To prove the existence of S we first s choose L2 f R2 such that L2 ∩ H = L1 ∩ N ∩ H (L2 exists since N ∩ H = x is potent in R2). Next we proceed as above: consider the natural epimorphism −→ %%= ∩ ∗ ; ψ: R R R1/L1 N HL2/L2 R2/L2 % let M be a normal subgroup of R% of finite index which intersects trivially both −1 R1/L1 ∩N and R2/L2; put S = ψ (M ), then clearly S f R and S ∩R1 = L1 ∩N. This completes the proof of the claim, and proves part (a) of the theorem in this case.

Case 2. The element x is hyperbolic. Then x does not stabilize any vertex or edge of the graph Sabs(R). By Lemma 8.3.2, x also acts freely on the proﬁnite graph S(R).ForN f R, write RN = R1N/N ∗HN/N R2N/N; and let ϕN : R −→ RN denote the natural canonical map. Then

R = R N/N R N/N and R= lim R ; N 1 2 ←− N HN/N and S(R) = lim S(R ) ←− N

(see Lemma 8.3.3). We use the same notation for the natural extension of ϕN , x x ϕN : R −→ RN . For each N, put xN = ϕN (x) ∈ RN . Denote by S(R) and S(RN ) N the sets of ﬁxed points of S(R) and S(RN ) under the actions of x and xN respectively. Then S(R) x = lim S(R )xN . ←− N 15.5 Amalgamated Products and Products of Cyclic Subgroups 407

Since x acts freely on S(R) , one has that S(R) x =∅; therefore there exists some , xM =∅ , = M f R !such that S(RM ) (see Sect. 1.1), i.e., xM is hyperbolic in RM R1M/M HM/M R2M/M; and, in particular, xM has infinite order. Now, RM is a free-by-finite group (cf. Serre 1980, Proposition II.11), and so it is quasi-potent (see ∼ Lemma 15.1.3(b)). Therefore xM = Z. Since ϕM maps x onto xM , we infer ∼ that x = Z, and so ϕM sends x isomorphically onto xM (see Sect. 1.3). Note that since RM is free-by-finite, it is subgroup separable (see Theorem 11.2.2); hence xM ∩RM = xM . It follows that

xM =ϕM x ≤ ϕM x∩R ≤ xM ∩RM = xM , and so ϕM ( x) = ϕM ( x∩R); consequently x= x∩R, since ϕM is injective on x. Thus x is closed in the proﬁnite topology of R. This completes the proof of part (b) in this case. Now, from the quasi-potency of RM , there is some natural number k such k k that RM is xM -potent. We deduce that R is x -potent: indeed, if m is a nat- ∩ = km −1 ural number and M0 f RM with M0 xM xM , then ϕM (M0) f R and −1 ∩ = km ϕM (M0) x x . This proves part (a) in this case.

15.5 Amalgamated Products and Products of Cyclic Subgroups

In this section we show that property (c) of X (the product of two cyclic subgroups is closed in the proﬁnite topology) is preserved by taking amalgamated free products with cyclic amalgamation. First we need to sharpen Proposition 8.3.1(f) when the graph consists of just abs G one edge and two vertices, so that Π1 ( ,)is an amalgamated free product.

Lemma 15.5.1 Let R1 and R2 be residually finite abstract groups with a common cyclic subgroup H such that each Ri is quasi-potent and H is closed in the profinite topology of each Ri . Define R = R1 ∗H R2, and let C = c be a cyclic subgroup of R. Assume that c is hyperbolic with respect to its action on the standard tree abs S (R) of the amalgamated free product R = R1 ∗H R2. Let Lc be the Tits line associated with c, and consider the subgroup

N ={r ∈ R | rLc = Lc} of R. Then (a) N is closed in the profinite topology of R; (b) N is a polycyclic-by-finite group with Hirsch number at most 2; (c) the profinite topology of R induces on N its full profinite topology.

Proof We use the set-up and the notation of 15.3.3. By Proposition 15.1.4 the hypotheses of Proposition 8.3.1 are satisﬁed; so part (a) is a special case of Proposi- tion 8.3.1(f). 408 15 Conjugacy Separability in Amalgamated Products

Since c is hyperbolic, C is inﬁnite; and obviously C ≤ N. Consider the natural homomorphism

ϕ : N −→ Aut(Lc) that sends an element t ∈ N to the automorphism of Lc consisting of left multiplication by t. Then the kernel K of ϕ ﬁxes every edge of Lc, and so it is conjugate to a subgroup of H , according to the deﬁnition of Sabs(R); in particular, K is cyclic; say K = k. Recall that Lc is an oriented graph whose underlying geometric graph has the form ···• • • •···

An automorphism of Lc determines uniquely an automorphism of this geometric graph; the automorphism group of this geometric graph is the infinite dihedral group Z C2; therefore Aut(Lc) is a subgroup of an infinite dihedral group. Since C acts freely on Lc,wehaveK ∩C = 1; so N/K is an infinite subgroup of an infinite dihedral group, and hence N/K is either an infinite cyclic group or an infinite dihedral group. Therefore N is a polycyclic-by-finite group with Hirsch length at most 2; this proves (b). To prove (c) let x ∈ N be such that xK generates a maximal infinite cyclic subgroup of N/K;so x∩K = 1 and x acts freely on Lc. Then either xK coincides ∼ with N/K or N/K = xK C2 is infinite dihedral; therefore, if r ∈ N − x,k, then rxr−1 ≡ x−1 (mod K). Put M = x,k; then M is a subgroup of N of index at most 2 in N. Consider the centralizer A = CM (K) of K in M. Since Aut(K) is finite, A = xs,k, for some natural number s.SoA is an abelian group of finite index in N; furthermore A = xs×K. We claim that A = CR(A). Note first that CR(A) ≤ N,forlet1= a ∈ A ∩ C (such a exists since A has finite index in N and C is an infinite subgroup of N) and note that Lc is also the Tits line corresponding to a, and so it is the unique abs minimal a-invariant subtree of S (R).Now,ifz ∈ CR(A), then zLc is a minimal abs a-invariant subtree of S (R), therefore, zLc = Lc,soz ∈ N.Ifr ∈ N − M, then s −1 −s rx r ≡ x (mod K), as indicated above, so that r/∈ CR(A); hence CR(A) ≤ M. Thus CR(A) = CM (A) = A. This proves the claim. Next note that the centralizer of an element in a topological group is closed; so A is closed in the profinite topology of R. Since A has finite index in N, it is open in the topology of N induced by the profinite topology of R. Therefore, to complete the proof of (c), it suffices to show that the profinite topology of R induces on A its full profinite topology. Put y = xs ; then A = y×K. By Proposition 15.4.1 R is quasi-potent and so the closure of any cyclic group coincides with its profinite completion. We distinguish two cases. Assume first that K is finite. Since y has infinite order, y= y =∼ Z; hence y∩K = 1. Therefore A¯ = y×K =∼ y×Kˆ =∼ y×K = A.ˆ That is, the profinite topology of R induces on A its own full profinite topology. 15.5 Amalgamated Products and Products of Cyclic Subgroups 409

Now consider the case when K is infinite. Since y acts freely on Lc,as pointed out above, y acts freely on Sabs(R) (indeed, if y were to fix a vertex v abs of S (R) − Lc, then y would map the chain joining v to a vertex in Lc to a chain abs joining v to a different vertex of Lc, producing a circuit in the tree S (R), a contradiction). In other words, y is hyperbolic (and in fact Ly = Lc by the minimality of Lc as a y-invariant subtree). According to Lemma 8.3.2, y acts freely on S(R) ¯ and in particular on Lc. Since K fixes all elements of Lc, it follows that K fixes all ¯ ¯ ¯ elements of Lc. We deduce that y∩K = 1, and so A = y×K. Thus A¯ = y×K¯ =∼ y×Kˆ =∼ y×K = A.ˆ

Proposition 15.5.2 Assume that the abstract groups R1 and R2 are quasi-potent and have property (c) of 15.3.1 (the product of two cyclic subgroups of Ri is closed in the proﬁnite topology of Ri (i = 1, 2)), and let H be a common cyclic subgroup of R1 and R2. Then the amalgamated free product R = R1 ∗H R2 has property (c).

≤ ˆ ˆ = Proof By Proposition 15.1.4, R R and R R1 H¯ R2.LetC1,C2 be cyclic subgroups of R. One must show that C1C2 ∩R = C1C2, where C1C2 denote the closure ˆ of C1C2 in R (see Lemma 11.1.1(c)). Note that C1C2 = C1 C2.Letγ1 ∈ C1,γ2 ∈ C2 and assume that γ1γ2 = k ∈ R. We need to show that γ1γ2 ∈ C1C2.Ifγ1 ∈ C1, then γ2 ∈ R ∩ C2 = C2, and the result is proved. So from now on we assume that γ1 ∈/ C1 and γ2 ∈/ C2. Consider the standard abstract tree Sabs = Sabs(R) associated with the amalgamated free product decomposition R = R1 ∗H R2, and the standard proﬁnite tree S = S(R)ˆ associated with the amalgamated free proﬁnite product decomposition ˆ = abs ˆ R R1 H¯ R2. Then S (R) is a dense subgraph of S(R) (see 15.3.3). We consider three cases.

Case 1. C1 and C2 are conjugate in R to subgroups of R1 or R2.

Say that C1 is conjugate in R to a subgroup of R1; then we may assume that abs C1 ≤ R1. Consider the vertex v1 = 1R1 = 1R1 of S ⊆ S and observe that its abs ˆ stabilizer under the action of R on S is R1, while under the action of R on S is R1. Since C2 is conjugate to a subgroup of either R1 or R2, it must fix a vertex abs abs of S ;letw beavertexinS fixed by C2 such that its distance to v1 is minimal; note that C2 fixes w as a vertex of S. We use induction on the length & of the chain [v1,w] to prove that γ1γ2 ∈ C1C2.If& = 0, then both C1 and C2 are subgroups of R1 and the assertion follows from the facts that R1 has property (c) by hypothesis, the profinite topology of R1 coincides with the topology induced from the profinite topology of R and R1 is closed in R (see Proposition 15.1.4). Assume now that &>0 and that the result holds whenever γ%2 ∈ C2 with abs γ1γ%2 ∈ R and γ%2 fixes a vertex w˜ ∈ S such that [v1, w˜ ] has length smaller than &.Lete denote the edge of the chain [v1,w] which is incident with w.Ob- −1 = −1 −1 = abs serve that γ2 v1 (γ1γ2) v1 and γ2 w w are vertices of S ; hence the chain −1[ ]=[ −1 ] abs −1 abs γ2 v1,w γ2 v1,w is in S , and, in particular, γ2 e is an edge of S incident with w. Since Sabs is the graph of an amalgamated free product, all the 410 15 Conjugacy Separability in Amalgamated Products

∈ = −1 edges are in the same R-orbit; hence there exists a gw R with gwe γ2 e.We ˆ deduce that gw ∈ C2(R)e = C2 Re (see 15.3.3). Since Re is a conjugate of H ,it is cyclic. On the other hand Rw is a conjugate of either R1 or R2, and so it has property (c); since C2,Re ≤ Rw, one has gw ∈ C2 Re ∩ R = C2Re. Hence there ∈ ∈ = = = −1 exist c2 C2,ge Re such that gw c2ge. Therefore c2e gwe γ2 e.De- = ∈ = ∈ fine γ2 γ2c2; then γ1γ2 R. Observe that γ2 γ2c2 C2 and γ2 fixes e.De- [ ] note by w1 the other vertex of e; then γ2 fixes w1 and the length of v1,w1 is − & 1. We infer from the induction hypothesis that γ1γ2 is in C1C2, and thus so is = = −1 k γ1γ2 γ1γ2c2 , as desired.

Case 2. C2 is conjugate in R to a subgroup of R1 or R2,butC1 is not. abs Then C2 ﬁxes a vertex of S ,sayv, and so C2 ﬁxes v considered as a vertex abs of S; hence γ1v = γ1γ2v ∈ S .Letc1 be a generator of C1; then c1 is hyperbolic.

Let Lc1 be the Tits line associated with c1 (see Proposition 8.1.3). Choose a vertex [ ] [ ] w of Lc1 ; then γ1 v,w is a finite chain of S, and hence γ1w,w is also finite, since [v,γ1v] is finite. Therefore, from Proposition 8.3.1(e) and (c) we deduce that γ1 ∈ C1, a contradiction. Thus this case does not arise.

Case 3. Neither C1 nor C2 is conjugate in R to a subgroup of R1 or R2.

Let C1 = c1 and C2 = c2. Then both c1 and c2 are hyperbolic elements with abs abs respect to their action on S . Denote by Lc1 and Lc2 the Tits lines in S associated with c1 and c2, respectively (see Proposition 8.1.3). Let v be a vertex of Lc2 . Since γ2 ∈/ C2, the chain [v,γ2v] is infinite (see Proposition 8.3.1(e) and (c)). Note that [v,γ2v] is a subgraph of the profinite subtree [γ2v,kv]∪[v,kv] of S, and [γ2v,kv] is a subgraph of [v,γ2v]∪[v,kv]; since [v,kv] is finite, it follows that [v,γ2v] and [γ2v,kv] differ by at most a finite number of vertices and edges. Therefore [v,γ2v]∩[γ2v,kv] is infinite. Hence, it follows from Corollary 8.3.4 that [ ]= =[ ]∩[ ] v,γ2v Lc2 v,γ2v γ2v,kv . = ∩ Suppose Lc1 Lc2 . Then Lc1 Lc2 is finite: indeed, one has that either ∩ = ∩ = ∩ = Lc1 Lc2 Lc1 or Lc1 Lc2 Lc2 ;sayLc1 Lc2 Lc1 ; next observe that by the definition of Lc1 (see Proposition 8.1.3), it does not contain any proper infinite subtrees. abs Hence, the image of Lc2 in S /Lc1 has infinite diameter. Let T be the minimal abs ⊆ C1-invariant subtree of S containing kv. Then Lc1 T (see Proposition 8.1.3(b)). ⊆ = −1 Furthermore, Lc2 T , because T contains kv and γ2v γ1 kv, and so it contains [ ] γ2v,kv , which in turn contains Lc2 , as shown above. Let w beavertexofLc1 . = [ ] = [ ]∪[ ] Then Lc1 C1 w,c1w (see Proposition 8.1.3(c)) and T C1( w,c1w w,kv ). Therefore T = C1([w,c1w]∪[w,kv]).

Next we claim that Lc2 is also contained in T . To prove this claim it sufﬁces ¯ ∩ abs = ∩ abs = to prove that T S T , because we already know that Lc2 S Lc2 (see ¯ ¯ abs ¯ Proposition 8.3.1(b)). To see this let t ∈ T ∩ S ;sayt = αs, where α ∈ C1 and s ∈ [ ]∪[ ] ¯ w,c1w w,kv ; then t is at a ﬁnite distance from the element αw of αLc1 . Since 15.6 Amalgamated Products and Normalizers of Cyclic Subgroups 411

¯ ∈ abs ¯ abs ∩ t S , the only elements at a finite distance from t are those in S ; hence αLc1 abs ¯ S = ∅; consequently α ∈ C1 (see Proposition 8.3.1(c)), and therefore t ∈ T ;this proves T¯ ∩ Sabs = T and so the claim. Denote by & the length of [w,kv], and observe that every vertex of T is at a = [ ] distance at most & from Lc1 C1 w,c1w . So the diameter of the quotient graph abs T/Lc1 is at most 2&. Since the image of Lc2 in S /Lc1 is contained in T/Lc1 ,it follows that the diameter of that image is finite. This contradiction shows that in fact = one never has Lc1 Lc2 . = Therefore the only possible situation in this case is that Lc1 Lc2 . Then by Lemma 15.5.1, C1,C2 ≤ N ≤ R, where N is closed in the profinite topology of R, it is polycyclic-by-finite and its full profinite topology coincides with the topology induced by the profinite topology of R. According to Proposition 11.4.7, C1C2 is closed in the profinite topology of N, and thus of R, as needed.

15.6 Amalgamated Products and Normalizers of Cyclic Subgroups

In this section we study property (f) of X and under which conditions it is preserved when taking amalgamated free products amalgamating a cyclic subgroup.

Proposition 15.6.1 Let R1 and R2 be quasi-potent abstract groups having properties (f) and (d) of 15.3.1. Suppose that H is a common cyclic subgroup of R1 and R2, closed in their respective proﬁnite topologies, and let R = R1 ∗H R2. Then R has property (f): if γ ∈ Rˆ and r is an element of inﬁnite order of R such that γ rγ −1 = r, then γrγ−1 ∈{r, r −1}.

Proof We adopt the general set-up and notation of 15.3.3.

Case 1. The element r is not hyperbolic, i.e., r ﬁxes a vertex of Sabs.

This means that r is conjugate in R to an element of R1 or R2;sowemayassume that r is in R1 or R2,sayinR1,theR-stabilizer of the vertex v = 1R1 = 1R1 of abs S ⊆ S.Ifalsoγ ∈ R1 = R1, then the result follows from property (f) for R1. Assume then that γ/∈ R1. Observe that in the graph S the edges incident with v ¯ ¯ −1 have the form %H , where % ∈ R1. Therefore, by Theorem 7.1.4, r ∈ δHδ ,for ∈ ˆ ˆ ¯ ¯ −1 some δ R1, because the R-stabilizer R%H¯ of %H is %H% . So, by property (d) applied to R1, r is conjugate in R1 to an element of H , and therefore we may assume that r ∈ H . Hence by Proposition 15.2.4(b), N ˆ ( r) = N ( r) ¯ N ( r). Consider the R R1 H R2 natural homomorphism

: −→ ϕ NRˆ r Aut r . Let C denote the subgroup of Aut( r) of order 2 consisting of the identity automorphism and the automorphism that inverts r. It follows from property (f) for R1 412 15 Conjugacy Separability in Amalgamated Products and R that the images of N ( r) and N ( r) in Aut( r) are in C. Hence Im(ϕ) 2 R1 R2 is contained in C, as desired.

Case 2. The element r is hyperbolic, i.e., r acts freely on Sabs. abs Let Lr be the Tits line in S corresponding to r and let Lr be its closure in S (see Proposition 8.1.3). Since Lr is the unique minimal r-invariant subtree of S (see ∈ Proposition 8.3.1(b)), NRˆ ( r ) acts naturally on Lr (indeed, if x NRˆ ( r ), then xLr is also a minimal r-invariant profinite subtree of S, and so xLr = Lr ). Note that every automorphism of the graph Lr is also an automorphism of the undirected graph ···• • • •··· therefore Aut(Lr ) is a subgroup of the infinite dihedral group. Consider the natural homomorphism ρ : r−→Aut(Lr ) that sends an element x of r to the automorphism ρx of the graph Lr consisting of left multiplication by x. Observe that ρ is injective since r acts freely on Lr (see Proposition 8.1.3). We identify r with its image in Aut(Lr ). Denote by A the subgroup of Aut(Lr ) consisting of all the automorphisms of Lr that extend to a continuous automorphism of the profinite graph Lr . Then we have

r≤A ≤ Aut(Lr ), where Aut(Lr ) is the group of continuous automorphisms of Lr . ˜ : −→ Consider now the natural homomorphism ρ NRˆ ( r ) Aut(Lr ) defined anal- ˜ ogously: an element x of NRˆ ( r ) is sent to the automorphism ρx of Lr consisting of left multiplication by x. Note that ρ˜ is injective when restricted to r since this group acts freely on Lr (see Lemma 8.3.2). ∈ ∈ By hypothesis γ NRˆ ( r ). Choose v Lr . By Proposition 8.3.1(a) there exists an r ∈ r such that r γv∈ Lr .Ifw ∈ Lr , the chain [v,w] is finite; so [r γv,r γw] is finite. Since r γw∈ Lr , the point r γwis in the same abstract connected component of the graph Lr as r γv; therefore r γw∈ Lr (see Proposition 8.3.1(e)). Thus r γ acts in fact on Lr .Soρ˜r γ ∈ A. Since A is a subgroup of the infinite dihedral group, ρ˜r γ normalizes any infinite subgroup of A, in particular it normalizes r. − − − ˜ 1 ˜ =˜ 1 ˜ ˜ 1 ˜ So ρr γ rρr γ ρr γ ρr ρr γ is either r or r . Since ρ is injective on r and since r and r commute, we deduce that (r γ)−1r(r γ)= γ −1rγ is either r or r−1,as desired.

15.7 Amalgamated Products and Intersections of Cyclic Subgroups

Here we show that property (e) of X (C1 ∩ C2 = 1 if and only if C1 ∩ C2 = 1, for cyclic C1 and C2) is preserved by amalgamated free products with cyclic amalgamation. First we indicate in the following lemma an equivalent way of stating property (e). 15.7 Amalgamated Products and Intersections of Cyclic Subgroups 413

Lemma 15.7.1 Let R be an abstract group such that (i) every cyclic subgroup of R is closed in the profinite topology of R (i.e., R is cyclic subgroup separable); (ii) the profinite topology of R induces on any cyclic subgroup its own full profinite topology.

Let C1 and C2 be cyclic subgroups of R. Then the following conditions are equivalent:

(a) C1 ∩ C2 = C1 ∩ C2. (b) C1 ∩ C2 = 1 if and only if C1 ∩ C2 = 1.

Proof If (a) holds, then obviously so does (b). Conversely, assume that (b) holds. If C1 ∩ C2 = 1, then certainly C1 ∩ C2 = 1, and so (a) holds. So from now on assume in addition that C1 ∩ C2 = 1. By (b), C1 ∩ C2 = 1. Hence, since C1 is cyclic, C1 ∩ C2 has ﬁnite index in C1, and so C1 ∩ C2 is an open subgroup of C1. Therefore K = C1 ∩ C2 is open in C1;so K ∩ C1 = K, since C1 = C1, according to property (ii). By property (i), Ci ∩R = Ci (i = 1, 2). Then

C1 ∩ C2 = C1 ∩ R ∩ C2 ∩ R = K ∩ C1 ∩ R = K ∩ C1.

Thus C1 ∩ C2 = K ∩ C1 = K = C1 ∩ C2, proving (a).

Proposition 15.7.2 Let R1 and R2 be quasi-potent, cyclic subgroup separable abstract groups having property (e) of 15.3.1. Suppose that H is a common cyclic subgroup of R1 and R2, and let R = R1 ∗H R2. Then for any pair of cyclic subgroups C1 and C2 of R, we have

C1 ∩ C2 = C1 ∩ C2 (in other words, R also satisﬁes property (e)).

Proof By Proposition 15.4.1 the group R satisfies conditions (i) and (ii) of Lemma 15.7.1; hence, according to this lemma, proving the present proposition is equivalent to proving that for any pair of cyclic subgroups C1 and C2 of R, C1 ∩ C2 = 1 if and only if C1 ∩ C2 = 1. We shall do this. If C1 ∩ C2 = 1, then certainly C1 ∩ C2 = 1. Therefore from now on we shall assume that C1 ∩ C2 = 1, and we need to prove that C1 ∩ C2 = 1. This is clear if C1 (and hence C2) is finite. So from now on we assume that both C1 and C2 are infinite. We adopt the general set-up and notation of 15.3.3. Say C1 = c1 and C2 = c2. According to Lemma 8.3.2, if an element a of R acts freely on Sabs (i.e., if a is hyperbolic), then a acts freely on S as well. Conse- quently, since C1 ∩ C2 = 1, we must have that c1 and c2 are either both hyperbolic or both nonhyperbolic. 414 15 Conjugacy Separability in Amalgamated Products

Case 1. Both c1 and c2 are nonhyperbolic. abs This means that ci fixes a vertex of S , i.e., ci is conjugate in R to an element of R1 or R2 (i = 1, 2).Let& be the minimal distance between two vertices u1 and abs u2 of S such that ui is fixed by ci (i = 1, 2). We shall prove that C1 ∩ C2 = 1 ∈ −1 ∈ −1 using induction on &. Say c1 r1R1r1 ,forsomer1 R. Replacing ci with r1 cir1 abs (i = 1, 2), we may assume that c1 ∈ R1. Then C1 fixes the vertex v1 = 1R1 of S . abs Let v denote a vertex of S fixed by c2 and closest to v1. Then & is the length of the chain [v1,v].If& = 0, then v1 = v and C1,C2 ≤ R1, so that the result follows by property (e) applied to R1. = ∈ −1 Next we consider the case & 1 separately. This means that c2 r1R2r1 ,for ∈ −1 = some r1 R1. Replacing ci with r1 cir1 (i 1, 2), we may assume in addition that c2 ∈ R2. Put v2 = 1R2 and e = 1H (regarding v2 and e as elements of R/R2 and R/H, respectively); then v = v2, and C1 ∩ C2 stabilizes v1 and v2 (as vertices of S), ¯ ¯ and therefore e. If follows that C1 ∩ C2 ≤ H .SoCi ∩ H ≥ C1 ∩ C2 = 1 (i = 1, 2). Applying property (e) to R1 and to R2 we get that C1 ∩ H = 1 = C2 ∩ H , and in particular H is infinite cyclic. It follows that C1 ∩ C2 = 1, as needed.

Assume now that &>1 and that the result holds whenever c1 and c2 are nonhyperbolic elements that ﬁx vertices of Sabs which are at a distance smaller than & and ∩ = [ ] ˜ such that c1 c2 1. Then the chain v1,v contains at least two edges. Let e ˜ be the edge of [v1,v] one of whose vertices is v1, and let e˜ be the edge of [v1,v] contiguous with e˜:

e˜ e˜ v1 • • ··· • v ˜ =˜ ˜ ∈ ˜−1 ˜ = One has that e r1e,forsomer1 R1. Replacing ci with r1 cir1 (i 1, 2), v with ˜−1 [ ] ˜−1[ ]=[ ˜−1 ] r1 v and v1,v with r1 v1,v v1, r1 v , we may assume that the ﬁrst edge of ˜ [v1,v] is e. Then the second vertex of [v1,v] is v2 and so e˜ = r2e,forsomer2 ∈ R2:

e r2e v1 v2 • ··· • v

Note that the group C1 ∩ C2 fixes the vertices and edges of [v1,v] (since it fixes v1 and v, and since S is a profinite tree which therefore does not contain finite cycles), and in particular it stabilizes e and r2e. Hence = ∩ ≤ ¯ ∩ ¯ −1 1 C1 C2 H r2Hr2 . ¯ ∩ ¯ −1 = ∩ −1 By property (e) of R2 and Lemma 15.7.1 we have that H r2Hr2 H r2Hr2 , and so = ∩ ≤ ∩ −1 1 C1 C2 H r2Hr2 . (15.4) ∩ −1 ∩ = [ ] − Therefore H r2Hr2 C2 1. Since the length of r2v1,v is & 2 and since ∩ −1 H r2Hr2 fixes r2v1 and C2 fixes v, it follows from the induction hypothesis that ∩ −1 ∩ = ∩ −1 H r2Hr2 C2 1. Since C2 is infinite, so is H r2Hr2 .Againfrom(15.4) ∩ −1 ∩ = ∩ −1 we obtain H r2Hr2 C1 1. Since H r2Hr2 also fixes v2 and C1 fixes v1, 15.8 Amalgamated Products and Conjugacy Distinguishedness 415

= ∩ −1 ∩ = we deduce from the case & 1 considered above that H r2Hr2 C1 1. Thus, ∩ = ∩ −1 C1 C2 1, since H r2Hr2 is inﬁnite cyclic.

Case 2. Both c1 and c2 are hyperbolic. = abs By Lemma 8.3.2, Ci acts freely on S(i 1, 2).LetLci be the Tits line of S = corresponding to ci (i 1, 2). By Proposition 8.3.1(b), Lci is the unique minimal ci-invariant profinite subtree of S(i= 1, 2). Since C1 ∩ C2 = 1, we have that C1 ∩ C2 is infinite. Since C1 ∩ C2 acts freely on S, there exists a unique minimal ∩ C1 C2-invariant subtree of S and therefore we have that Lci is also the unique minimal C1 ∩ C2-invariant subtree of S(i= 1, 2) (see Proposition 2.4.12(b) and (c)). = Hence Lc1 Lc2 . Then, according to Proposition 8.3.1(b), = ∩ abs = ∩ abs = Lc1 Lc1 S Lc2 S Lc2 . = ∈ ∪ So xLc1 Lc1 , for every x C1 C2. Therefore according to Lemma 15.5.1, there exists a polycyclic subgroup N of R containing C1 and C2 such that N is closed ¯ ˆ ˆ in the profinite topology of R and N = N. Then Ci (the closure in R) coincides ¯ ˆ with the closure of Ci in N = N. But for polycyclic groups C1 ∩ C2 = C1 ∩ C2 (see Proposition 13.2.8). Since by hypothesis C1 ∩ C2 = 1, we deduce that C1 ∩ C2 = 1, as needed.

15.8 Amalgamated Products and Conjugacy Distinguishedness

In this section we prove that property (d) of X (conjugacy distinguishedness of cyclic subgroups) is preserved by forming amalgamated free products with amalgamated cyclic subgroups, under some conditions. In contrast with the other properties that we have studied in Sects. 15.5Ð15.7, it appears that in this case one needs to assume in fact that the factors involved belong to the class X , not merely that they satisfy only some of the properties (a)Ð(f) that deﬁne X .

Proposition 15.8.1 Let R1 and R2 be groups in X with a common cyclic subgroup H . Then every cyclic subgroup of R = R1 ∗H R2 is conjugacy distinguished, ˆ i.e., if a,c ∈ R and aR ∩ c=∅, then aR ∩ c=∅.

Proof Assume that cz = γaγ−1, where γ ∈ Rˆ and z ∈ Zˆ . We need to show the existence of some g ∈ R such that gag−1 ∈ c. Note that we may assume that a (and so c) has infinite order because when the order of c is finite, c= c, and the result follows from Theorem 15.3.4. ˆ = = By Proposition 15.1.4, R R1 Hˆ R2 R1 H¯ R2. Consider the standard abstract tree Sabs = Sabs(R) (respectively, profinite tree S = S(R)ˆ ) associated with the amalgamated free product R = R1 ∗H R2 (respectively, the amalgamated free profi- ˆ = nite product R R1 H¯ R2). We adopt the general set-up and notation of 15.3.3. 416 15 Conjugacy Separability in Amalgamated Products

Case 1. The element a is not hyperbolic, i.e., a fixes a vertex of Sabs. First we note that then c fixes a vertex of Sabs, for otherwise, according to Lemma 8.3.2, c, and hence a, would act freely on S, contradicting our hypothesis. This means that a and c are conjugate in R to elements of R1 or R2;sowe −1 may assume that a ∈ R1 ∪ R2 and rcr = c ∈ R1 ∪ R2,forsomer ∈ R. Then (c )z = rγaγ−1r−1.Now,aR ∩ c=∅if and only if aR ∩ c =∅. Thus replacing c with c and γ with rγ, we may assume that c ∈ R1 ∪ R2. So from now on we assume that a ∈ R1 ∪ R2 and c ∈ R1 ∪ R2. Say, a ∈ Rj , where j ∈{1, 2}. If, in addition, γ ∈ Rj = Rj and c ∈ Rj , then the result follows from property (d) applied to Rj . Note that since c ∈ R1 ∪ R2, the condition c ∈ Rj is equivalent to the condition c ∈ Rj , because (see Proposition 15.1.4) Rj is closed in the profinite topology of R, so that R ∩ Rj = Rj . If, on the other hand, either γ/∈ Rj or c/∈ Rj , then it follows from Corol- ¯ −1 −1 ¯ lary 7.1.5(b) that a ∈ βHβ ,forsomeβ ∈ Rj , and so β aβ ∈ H . Since H is cyclic, one can apply property (d) of Rj to get that for some b ∈ Rj we have b−1ab ∈ H . Therefore replacing a with b−1ab, we may assume from now on that a ∈ H . Hence either a ∈ H and c ∈ R1 or a ∈ H and c ∈ R2. For definiteness, from now on we shall assume that a ∈ H , c ∈ R1 and γ/∈ R1. abs Consider the vertices v1 = 1R1 = 1R1, v2 = 1R2 = 1R2 of S (and of S), and denote by e the edge in Sabs (and in S) joining these two vertices: e = 1H = 1H¯ . z −1 z Note that since c = γaγ ∈ R1, the element c fixes the distinct vertices v1 and z γv1 of S. In particular, the subset T of S fixed by c is nonempty. Hence by Theo- rem 4.1.5(a), T is in fact a profinite subtree of S. Since this subtree contains the two different vertices v1 and γv1, there exists an edge e1 ∈ T one of whose vertices is v1 (see Proposition 2.1.6(c)). Since, by definition, the set of edges of S is R/ˆ H¯ , there ˆ is a γ1 ∈ R such that γ1e1 = e; but since v1 is also a vertex of e, we must have that ˆ z γ1 is in R1, which is the stabilizer of v1 in R. Now, since e1 ∈ T ,wehavec e1 = e1; z −1 = z −1 ∈ ¯ ˆ so γ1c γ1 e e; hence γ1c γ1 H , the stabilizer of e in R. Next observe that a z −1 ¯ ˆ and γ1c γ1 are elements of the procyclic subgroup H and they are conjugate in R; = z −1 −1 ∈ therefore, according to Lemma 15.2.1, a γ1 c γ1 .Soγ1 aγ1 c ; thus by property (d) applied to R1, we have that a is conjugate in R1 to an element of c, as desired.

Case 2. The element a is hyperbolic, i.e., a does not ﬁx any vertex of Sabs. Then by Lemma 8.3.2 a acts freely on S;so cz=γ aγ −1 acts freely on S, and therefore c must act freely on S, and in particular on Sabs; i.e., c is hyperbolic. Let abs La and Lc be the Tits lines of S corresponding to a and c, respectively. Choose v ∈ La and w ∈ Lc. Then

La = a[v,av] and Lc = c[w,cw]. Consider the edge e = 1H = 1H¯ of Sabs ⊆ S. We claim that we may assume that e ∈[v,av]∩[w,cw]. To see this let ra,rc ∈ R be such that e ∈ ra[v,av] and e ∈ 15.8 Amalgamated Products and Conjugacy Distinguishedness 417

[ ] = −1 = −1 rc w,cw . Deﬁne a1 raara , c1 rccrc , and observe that a1 and c1 are also hyperbolic elements of R and

−1 −1 −1 = z rcγra a1 rcγra c1. abs Let La1 and Lc1 be the Tits lines of S corresponding to a1 and c1, respectively. Then (see Lemma 8.1.6) = = [ ] La1 raLa a1 v1,a1v1 , = ∈ ∈[ ] = = where v1 rav La1 ; moreover, e v1,a1v1 . Similarly, Lc1 rcLc [ ] = ∈ ∈[ ] R ∩ =∅ c1 w1,c1w1 , where w1 rcw Lc1 ; and e w1,c1w1 . Since a c R ∩ =∅ −1 if and only if a1 c1 , after replacing a with a1, c with c1 and γ with rcγra , the claim is proved. Next we show that we may assume that γ ∈ H¯ . By Proposition 8.3.1(b), La = a[v,av] is the unique minimal a-invariant profinite subtree of S, and Lc = c[w,cw] is the unique minimal c-invariant profinite subtree of S. Since z −1 z c = γaγ , it follows that γ La is the unique minimal c -invariant profinite z subtree of S; therefore γ La ⊆ Lc,for c acts on Lc. We show now that in fact γ La = Lc. By the minimality of Lc as a c-invariant profinite subtree, it suffices to z show that γ La is c-invariant. To prove this, let c˜ ∈ c, and observe that c acts z z on cγ˜ La, since c and c˜ commute; so cγ˜ La is a minimal c -invariant profinite subtree of S, and therefore cγ˜ La = γ La, as desired. From γ La = Lc = c[w,cw] abs we deduce that γe∈ Lc. Choose c ∈ c such that c γe∈[w,cw]⊆S . Then c γe= re,forsomer ∈ R, because all the edges of Sabs are in Re. Since the stabilizer of e ∈ S under the action of Rˆ is H¯ , there exists some δ ∈ H¯ such that c γ = rδ. Now, a = γ −1c −1czc γ = δ−1r−1czrδ. Therefore, using r−1 cr instead of c, we may assume that γ is in H¯ , as asserted. By Proposition 15.1.4, = ∗ ≤ ∗ ≤ = ˆ R R1 H R2 R1 H¯ R2 R1 H¯ R2 R. Since a ∈ R and γ ∈ H¯ ,wehave z = −1 ∈ ∗ c γaγ R1 H¯ R2. z z So c can be written as a product c = w1w2 ···wm, where wi ∈ R1 ∪ R2 (i = 1,...,m). This means that the chain [e,cze] of S is finite. As indicated above z e ∈ Lc, and note that [e,c e]⊆Lc. By Proposition 8.3.1(e), Lc is a connected com- z z ponent of Lc, considered as an abstract graph. Hence c e ∈ Lc. Therefore, c ∈ c, according to Proposition 8.3.1(c). So γaγ−1 = cz ∈ c≤R. Now, it follows from Theorem 15.3.4 that there exists some g ∈ R such that gag−1 = cz ∈ c, as needed. 418 15 Conjugacy Separability in Amalgamated Products

15.9 Conjugacy Separability of Certain Iterated Amalgamated Products

From Theorem 15.3.4 and Propositions 15.5.2Ð15.8.1 one deduces the following theorem.

Theorem 15.9.1 Let R1,R2 ∈ X . Then their amalgamated free product R1 ∗H R2 amalgamating a common cyclic subgroup H is in X .

As pointed out in Example 15.3.2, free groups, free abelian groups, finitely generated free-by-finite groups and polycyclic-by-finite groups are in the class X . These examples together with Theorem 15.9.1 provide a large source of conjugacy separable groups, as we make explicit in the following result.

Theorem 15.9.2 Let X1 be the class of all abstract groups that are either free or free abelian or finitely generated free-by-finite or polycyclic-by-finite. For i>1, define recursively the class Xi to consist of all groups that are amalgamated free products

R = R1 ∗H R2 of groups R1,R2 in Xi−1 amalgamating a common cyclic subgroup H . Then any group in the class ∞

X = Xi i=1 is conjugacy separable.

15.10 Examples of Conjugacy Separable Groups

Example 15.10.1 (Surface groups) The fundamental group of a ﬁnite surface is called a surface group. Depending on the genus g, the number of boundary components r and the orientability of the surface, these groups admit one of the following presentations (cf. Zieschang, Vogt and Coldewey 1980, Theorem 3.2.8): Ð in the case of an orientable surface, s1,...,sr ,t1,u1,...,tg,ug s1 ···sr [t1,u1]···[tg,ug] ; and Ð in the case of a non-orientable surface, ··· 2 ··· 2 s1,...,sr ,v1,...,vg s1 sr v1 vg . Therefore, a surface group is ﬁnitely generated and either abelian or an amalgamated free product of two free groups amalgamating a cyclic subgroup. Thus, using Theorem 15.9.2, one has the following proposition. 15.10 Examples of Conjugacy Separable Groups 419

Proposition 15.10.2 Surface groups are conjugacy separable.

Example 15.10.3 Lyndon group Let A be a ring with identity 1. An abstract group R is called an A-group if it comes equipped with a function R × A → R, (g,α) → gα satisfying the following conditions (α, β ∈ A,g,h ∈ R): g1 = g, gαgβ = gα+β , gαβ = (gα)β , (g−1hg)α = g−1hαg, and if g and h commute, then (gh)α = gαhα. Every group R is naturally a Z-group; and there is a standard way of ‘enlarg- ing’ R to a ‘minimal’ A-group RA (cf. Myasnikov and Remeslennikov 1994, 1996, for the precise deﬁnition and construction of RA, where it is called the tensor A- completion of R). When R is abelian, the group RA is just the ordinary tensor product R ⊗ A over Z. In general, R is not a subgroup of RA. When R = Φ is a free abstract group and A = Z[t] is the ring of polynomials in one indeterminate with integer coefﬁcients, the group ΦZ[t] was originally studied by R. Lyndon (cf. Lyn- don 1960) and it receives now the name Lyndon group. In this case Φ ≤ ΦZ[t]. Myasnikov and Remeslennikov (1996) give an explicit construction of the Lyndon group in terms of amalgamated free products of groups with cyclic amalgamations as follows: 1st step: One starts with the group Φ(0) = Φ. 2nd step: We consider a tree of groups of the form

C1 ⊗ Z[t] C1

Φ(0) C2 ⊗ Z[t] C2 . Ci . .

Ci ⊗ Z[t] (0) where {Ci | 1 ≤ i ≤ δ0} is a collection of inﬁnite cyclic subgroups of Φ indexed by the ordinals less than or equal a certain ordinal number δ0 (more precisely, the Ci are representatives of the conjugacy classes of all centralizers of nontrivial elements of Φ(0) = Φ, which of course in this case are all of them maximal cyclic subgroups). The edge group Ci is embedded into the vertex group Ci ⊗ Z[t] by the map Ci → Ci ⊗ Z[t] that sends c ∈ Ci to c ⊗ 1; this is indeed an embedding because Ci is inﬁnite cyclic. Let Φ(1) be the fundamental group (the tree product) of this graph of groups. Then Φ(1) is the union of a chain Φ = Φ(0) = Φ(00) ≤ Φ(01) ≤···≤···≤Φ(0i) ≤···≤Φ(0δ0) = Φ(1), where each i is an ordinal, 1 ≤ i ≤ δ0, and if i ≥ 1 is not a limit ordinal, then (0i) = (0 i−1) ∗ ⊗ [ ] (0i) = (0j) Φ Φ Ci Ci Z t , while if i is a limit ordinal, then Φ j

Then the Lyndon group is ∞ ΦZ[t] = Φ(m) m=0 (cf. Myasnikov and Remeslennikov 1996, Theorem 8). Therefore one can describe ΦZ[t] as a union of a chain of groups Z[t] Φ = Φ(0) ≤ Φ(1) ≤···≤Φ(i) ≤···≤Φ(δ) = Φ indexed by the ordinals i less than or equal to a certain ordinal δ and such that = ∗ ⊗ [ ] Φ(i) Φ(i−1) Ci (Ci Z t ), when i is a nonlimit ordinal, while if i is a limit = ordinal, then Φ(i) j

Lemma 15.10.4 Let 0 ≤ i ≤ s ≤ δ. Then

(a) there exists a group epimorphism ϕs,i : Φ(s) → Φ(i) which is the identity on the subgroup Φ(i) of Φ(s); (b) Φ(s) = Ks,i Φ(i), for some normal subgroup Ks,i of Φ(s).

Proof (a) We shall use transfinite induction to define a group homomorphism ϕs,i from Φ(s) onto Φ(i) which is the identity on the subgroup Φ(i) of Φ(s), and such that the restriction of ϕs,i to Φ(k) is ϕk,i whenever i ≤ k ≤ s. Define ϕi,i to be the identity map and assume that ϕr,i has already been defined for all r

Proposition 15.10.5 The Lyndon group ΦZ[t] is in the class X (see 15.3.1), and in particular it is conjugacy separable.

Proof We continue with the above notation. We shall prove inductively that in fact each Φ(s) is in the class X , for all 0 ≤ s ≤ δ. Since Φ(0) = Φ is free, it is in class X . Assume that Φ(j) ∈ X for 0 ≤ j

−1 −1 ¯ have that γ˜ aγ˜ is either a or a . Since ϕ,s,i is the identity on A, we deduce that also γ −1aγ ∈{a,a−1}.

Example 15.10.6 (Fuchsian groups) A Fuchsian group is a finitely generated group of Möbius transformations αz + β z → z = (α,β,γ,δ∈ R,αδ− βγ = 1) γz+ δ of the extended complex plane which map a circular disk onto itself, or equivalently, a finitely generated discontinuous group of orientation-preserving motions of the hyperbolic plane. According to a theorem of Fricke (cf. Magnus 1974, p. 98), a Fuchsian group admits a presentation of the form − − = e1 en 1 ··· 1[ ]···[ ] F a1,b1 ...,ag,bg,c1,...,cm c1 ,...,cn ,cm c1 a1,b1 ag,bg , where 0 ≤ n ≤ m,0≤ g and the exponents ei are natural numbers ≥ 2. If m−n>0, applying a Tietze transformation (cf. Magnus, Karrass and Solitar 1966, p. 50), the presentation of F can be changed to = e1 en F a1,b1 ...,ag,bg,c1,...,cm−1 c1 ,...,cn , which is the presentation of a free product of finitely many cyclic groups; hence in this case F is conjugacy separable according to Corollary 14.1.2. Therefore we may assume that F has a presentation of the form − − = e1 en 1 ··· 1[ ]···[ ] F a1,b1 ...,ag,bg,c1,...,cn c1 ,...,cn ,cn c1 a1,b1 ag,bg . Except in two cases ((1) g = 0, n = 3, and (2) g = 1, n = 1) these groups can be expressed as an amalgamated free product with cyclic amalgamation of two groups that are free products of cyclic groups, and so one can use Theorem 15.9.2 to verify that they are conjugacy separable. The two exceptional cases correspond to the groups = e1 e2 e3 = r s t F c1,c2,c3 c1 ,c2 ,c3 ,c1c2c3 a,b a ,b ,(ab) and = e1 −1[ ] = [ ]e F a1,b1,c1 c1 ,c1 a1,b1 a,b a,b . These two special cases are treated in Stebe (1972) and Fine and Rosenberger (1990).

15.11 Notes, Comments and Further Reading: Part III

The properties of ‘residual finiteness’, ‘subgroup separability’, ‘conjugacy separability’, etc., are very powerful and few groups are known to satisfy one of these properties. If in addition the group in question is finitely presented, then one can prove that there are algorithms to answer some of the corresponding classical questions in combinatorial group theory. For example, if an abstract group R is finitely 15.11 Notes, Comments and Further Reading: Part III 423 presented and subgroup separable, then there is an algorithm to decide whether or not a given element of R belongs to a given finitely generated subgroup of R (the generalized word problem). This was first proved in Mal’cev (1958), who credits McKinsey (1943) with basic results. See also Mostowski (1966). Theorem 11.2.2 and Theorem 11.2.3 for free groups are contained in Hall (1949), Theorem 5.1; Theorem 11.2.3 is made explicit in Burns (1969). The proof that we present here for Theorem 11.2.2 is due to Stallings (1983). Theorem 11.2.4 and Corollary 11.2.5 are proved in Ribes and Zalesskii (1994). Theorem 11.3.8 was proved in Ribes and Zalesskii (1993). For the case of two factors it was proved in Niblo (1992); see also Gitik and Rips (1995). The more general Theorem 11.3.6 was proved in Ribes and Zalesskii (1994). Theorem 11.3.8 was conjectured in Pin and Reutenauer (1991); in fact they showed in that paper that The- orem 11.3.8 would also solve a question in finite semigroups, the so-called Rhodes conjecture (see Sect. 12.3 for additional information). Several other proofs of The- orem 11.3.8 have been given: Herwig and Lascar (2000) (using model-theoretic and combinatorial methods); Auinger (2004), Auinger and Steinberg (2005)(using inverse automata); Steinberg (2002) (using inverse monoid theory) and Auinger and Steinberg (2004), where Theorem 11.3.6 is extended to ‘arboreous’ pseudovarieties C. See also Rhodes and Steinberg (2009), Chap. 4, where there is a self- contained account of Theorem 11.3.8 and how it implies the Rhodes conjecture.

Subgroup Separability and Free Products

We collect here some results on the free product of subgroup separable groups and n-product subgroup separable groups.

Theorem 15.11.1 (Romanovskii 1969; Burns 1971) The free product of subgroup separable groups is subgroup separable.

The next result generalizes Theorem 11.3.8 and Theorem 15.11.1;itbuildson the combinatorial and model-theoretic methods of Herwig and Lascar (2000) and on Gitik (1997).

Theorem 15.11.2 (Coulbois 2000, 2001) Let n be a fixed natural number, and assume that Ri (i = 1, 2) is an abstract group such that whenever H1, ..., Hn are finitely generated subgroups of Ri , then the product H1 ···Hn is a closed subset in the profinite topology of Ri . Then the free product R1 ∗ R2 satisfies the same property.

If instead of the proﬁnite topology one considers the pro-C topology, then only the following partial generalization of Theorem 11.3.6 and Theorem 15.11.2 is known. Its proof is also based on the theory of abstract groups acting on abstract trees and on the theory of pro-C groups acting on C-trees, using ideas similar to those developed in the proof of Theorem 11.3.6. 424 15 Conjugacy Separability in Amalgamated Products

Theorem 15.11.3 (Ribes and Zalesskii 2004) Let C be an extension-closed pseudovariety of ﬁnite groups. Assume that the abstract groups R1 and R2 are 2-product subgroup separable (with respect to the pseudovariety C), i.e., in each Ri (i = 1, 2), the product of any two ﬁnitely generated closed subgroups in the pro-C topology is a closed subset. Then the same property holds in their free product R1 ∗ R2.

Open Question 15.11.4 Let C be an extension-closed pseudovariety of ﬁnite groups and let n be a natural number, n ≥ 3. Assume that the abstract groups R1 and R2 are n-product subgroup separable (with respect to the pseudovariety C), i.e., in each Ri (i = 1, 2), the product of any n ﬁnitely generated closed subgroups in the pro-C topology is a closed subset. Then does the same property hold in their free product R1 ∗ R2?

Another extension of Theorem 11.3.8 is included the following result.

Theorem 15.11.5 (You 1996) Let R = Φ × Z, where Φ is a free abstract group endowed with its proﬁnite topology. If H1,...,Hn are ﬁnitely generated subgroups of R, where n is any natural number, then their product H1 ···Hn is a closed subset of R.

Open Question 15.11.6 Let C be an extension-closed pseudovariety of ﬁnite groups (other than the class of all ﬁnite groups). Let R = Φ × Z, where Φ is a free abstract group. Is Rn-product subgroup separable (with respect to C)?

For possible extensions of Theorem 15.11.5, we note that, in general, the direct product of free groups is not subgroup separable (cf. Allenby and Gregorac 1973). Conjugacy separability as well as conjugacy distinguishedness are preserved by taking direct products. For an example of a subgroup separable group which is not conjugacy separable see Chagas, de Oliveira and Zalesskii (2012). An amalgamated free product of two subgroup separable groups need not be subgroup separable, even if the amalgamated subgroup is cyclic (cf. Rips 1990, and more explicitly Allenby and Doniz 1996). However one has the following result (see also Scott 1978; Allenby and Tang 1993; Gitik 1997;Wise2000, 2006, where the approach is very different from the methods used in this book).

Theorem 15.11.7 (Brunner, Burns and Solitar 1984) The amalgamated free product R = Φ1 ∗C Φ2 of two free groups Φ1 and Φ2 amalgamating a cyclic subgroup C is subgroup separable.

Open Question 15.11.8 Let R be an amalgamated free product of two free groups amalgamating a cyclic subgroup. Is Rn-product subgroup separable (n ≥ 2): if H1,...,Hn are ﬁnitely generated subgroups of R, is the set H1 ···Hn closed in the proﬁnite topology of R?

Observe that Open Question 15.11.8 is partially answered in Proposition 15.5.2. 15.11 Notes, Comments and Further Reading: Part III 425

Open Question 15.11.9 (Tang) Are one-relator groups with torsion subgroup separable?

For other amalgamated free products with not necessarily cyclic amalgamation see Gitik and Rips (1995), Kim and Tang (2002), Metaftsis and Raptis (2004), Zhou and Kim (2013). For subgroup separability in fundamental groups of graphs of groups see Raptis, Talelli and Varsos (1995). Theorem 11.4.1 was proved by Mal’cev (1958). Proposition 11.4.7 is due to J.C. Lennox and J.S. Wilson (cf. Lennox and Wilson 1979; see also Stebe 1976 for a proof of this proposition in the special case of finitely generated nilpotent groups). The proof presented here is suggested in Segal (1983), Chap. 4, Exercise 13. In Lennox and Wilson (1979) there is also an example of a polycyclic group R with three subgroups R1, R2 and R3 such that their product R1R2R3 is not closed in the profinite topology of R. For a different approach to some separability properties in groups using inverse monoids see Margolis, Sapir and Weil (2001) and Delgado, Margolis and Steinberg (2002). For some separability properties in 3-manifold groups see Aschenbrenner, Friedl and Wilton (2015). Algorithm 12.2.4 as well as Corollary 12.2.5, were obtained in Ribes and Zalesskii (1994) in answer to questions posed by J.-E. Pin and S. Margolis. Lemma 12.2.3 is due to Margolis, Sapir and Weil (2001); this paper approaches the existence of an algorithm to calculate the pro-p closure of a finitely generated group of a free group using the language of automata theory; it contains explicit examples and estimations. More examples can be found in Weil (2000). The framework in Sect. 12.3 uses ideas from Pin (1989) and Pin and Reutenauer (1991). For background and motivation for the results in Sect. 12.3 one may consult Pin (1989), Pin and Reutenauer (1991), Henckell, Margolis, Pin and Rhodes (1991), Almeida (1994), Steinberg (2001a,b), Rhodes and Steinberg (2009) (especially Chap. 4) and references therein. For partial implementations of computations of kernels K(M), see Delgado and Héam (2003). The conjecture of Rhodes concerning the existence of an algorithm to compute K(M) (see Theorem 12.3.13)was first confirmed by Ash (1991) using combinatorial and graph theoretic methods, very different from the ideas presented here. The sources for Sect. 13.1 are Ribes and Zalesskii (1996, 2014, 2016) and Ribes, Segal and Zalesskii (1998). In Theorem 13.1.7, the assumption that R is finitely generated is probably unnecessary. It is certainly not needed when R = Φ is a free group. Indeed, in that case one can write Φ = Φ1 ∗ Φ2, where Φ1 is a free group of finite rank and H ≤ Φ1; hence one can replace Φ with Φ1 and assume that R = Φ is finitely generated.

Open Question 15.11.10 Does one need ﬁnite generation of R in Theorem 13.1.7?

The approach that we have followed for the proof Theorem 13.1.7 depends on the fact that R can be interpreted as the fundamental group Πabs of a graph (G,)of finite groups over a finite graph ; and then relating Π abs with its completion, which 426 15 Conjugacy Separability in Amalgamated Products happens to coincide (because is finite) with the fundamental group of (G,) considered as a graph of pro-C groups. If R is not finitely generated, one can still interpret R as the fundamental group Π abs of a graph (G,)of finite groups, but then is no longer a finite graph (cf. Cohen 1973; Scott 1974). However it is not clear how to use this productively with the methods developed here. The most immediate difficulty is that is not a profinite graph. See Open Question 6.7.1. The results in Sect. 13.2 are taken mainly from Ribes, Segal and Zalesskii (1998). Sections 14.1 and 14.3 are based mostly on Ribes and Zalesskii (2014, 2016). Corollary 14.1.2 is proved in Stebe (1970) and in Remeslennikov (1971). Corol- lary 14.1.3 is attributed to Baumslag and Taylor in Lyndon and Schupp (1977), Proposition I.4.8. Theorem 14.1.4 was proved by Dyer (1979) in the special case when C is the pseudovariety of all finite groups, and by Toinet (2013) when C is the pseudovariety of all finite p-groups. Theorems 14.2.1 and 14.2.3 appear in Chagas and Zalesskii (2015).

Open Question 15.11.11 Does one need ﬁnite generation of R in Theorem 14.2.1 and Corollary 14.2.2?

Chapter 15 draws mainly from Ribes and Zalesskii (1996) and Ribes, Segal and Zalesskii (1998). The term ‘potent’ seems to have been first used in Allenby and Tang (1981), although the concept has an earlier history. Quasi-potent groups are sometimes referred to as ‘weakly-potent’ groups, see Tang (1995), where Lemma 15.1.3 is also proved. See Burillo and Martino (2006) where the class of quasi-potent groups whose cyclic subgroups are separable is studied. For a special case of Corollary 15.3.5, see Tang (1995). For studies of conjugacy separability in fundamental groups of graphs of groups, see Shirvani (1992) and Raptis, Talelli and Varsos (1998). Proposition 15.4.1(b) was first proved by Evans (1974). For a special case of Theorem 15.9.2, see Tang (1997). Proposition 15.10.5 is proved in Lioutikova (2003); the proof presented here is from Ribes and Zalesskii (2016). For analogues of limit groups in the context of pro-p groups, see Kochloukova and Za- lesskii (2011). In Kharlampovich and Myasnikov (1998) it is proved that every limit group is embeddable into a Lyndon group. A consequence of Theorem 11.2.3 is that every finitely generated subgroup H of an abstract free group Φ is a virtual retraction, i.e., there exists a subgroup U of finite index in Φ with H ≤ U such that there is an epimorphism ϕ : U → H which is the identity on H . This is often a useful property by itself.

Theorem 15.11.12 (Wilton 2008, Theorem B) Every ﬁnitely generated subgroup of a limit group is a virtual retraction.

Bogopolski and Bux (2015) show that surface groups are subgroup conjugacy separable. Chagas and Zalesskii (2016) extend this to limit groups. A Bianchi group is a group of the form PSL2(Od ), where d is a positive square-√ free integer and Od is the ring of integers of the quadratic number ﬁeld Q( −d) (cf. Bianchi 1892;Fine1989). The conjugacy separability of these groups is studied in Wilson and Zalesskii (1998) and Chagas and Zalesskii (2010). 15.11 Notes, Comments and Further Reading: Part III 427

Conjugacy Separability, Subgroups and Extensions

In Goryaga (1986) there is an example of a group which is not conjugacy separable but contains a subgroup of index two that is conjugacy separable. Also we know that a subgroup of ﬁnite index of a conjugacy separable group need not be conjugacy separable (cf. Chagas and Zalesskii 2009; Martino and Minasyan 2012). We state next a criterion to decide whether or not conjugacy separability is inherited by subgroups of ﬁnite index (cf. Chagas and Zalesskii 2009; Minasyan 2012, Proposi- tion 3.2).

Lemma 15.11.13 Let R be an abstract group. The following conditions are equivalent: (a) Every subgroup of ﬁnite index in R is conjugacy separable; = ∈ (b) R is conjugacy separable and CR(r) CRˆ (r), for every r R.

A group R satisfying the equivalent conditions of Lemma 15.11.13 is said to be hereditarily conjugacy separable; similarly, if C is a pseudovariety of ﬁnite groups, a conjugacy C-separable group R is said to be hereditarily conjugacy C-separable if every open subgroup of R is also conjugacy C-separable. See Ferov (2016), The- orem 4.2, for a generalization of Lemma 15.11.13 in the context of conjugacy C- separability. It is clear from Theorem 14.1.4 that free-by-C groups are hereditarily conjugacy C-separable (for extension-closed C); from Theorem 11.4.4 it follows that polycyclic-by-ﬁnite groups are hereditarily conjugacy separable. For other examples of hereditarily conjugacy separable groups, see Chagas and Zalesskii (2009, 2013), Martino and Minasyan (2012), and Minasyan (2012).

Conjugacy Distinguished Subgroups

Theorem 15.11.14 (Ribes and Zalesskii 2016) Let H be a ﬁnitely generated subgroup of an abstract group R. Then H is a conjugacy distinguished subgroup of R in each of the following cases: n (a) R is a one-relator group of the form R = x1,...,xn | W with n>|W|; (b) R = ΦZ[t] is a Lyndon group, where Φ is an arbitrary abstract free group; (c) R is a limit group.

The special case of Theorem 15.11.14(c) when R is a surface group also follows from Theorem 1.4 in Bogopolski and Bux (2015). Appendix A Abstract Graphs

The purpose of this Appendix is twofold. First we want to develop a terminology common to both abstract and proﬁnite graphs that is appropriate for this book. Every proﬁnite graph has the underlying structure of a graph in the abstract sense, if we dispense with the topology. The second purpose is to establish some basic results that are needed in parts of this work, using this common terminology. The choice of these results is dictated by our needs in the book.

A.1 The Fundamental Group of an Abstract Graph

An abstract (oriented) graph consists of a set together with a nonempty subset V = V()(the vertices of the graph) and two incidence maps

d0,d1 : −→ V whose restriction to V are the identity map on V .ThesetE = E() = − V is the set of edges of the graph. If e ∈ E, d0(e) is the initial vertex (or origin)ofe, and d1(e) is the terminal vertex (or terminus or end vertex) of e. A graph with only one edge e and one vertex v with d0(e) = d1(e) = v is called a loop based at v. A nonempty subset of is called a subgraph if whenever m ∈ , then d0(m), d1(m) ∈ ; observe that then is a graph in a natural way. A morphism (or a ‘map of abstract graphs’) α : −→ Λ of abstract graphs is a map such that di(α(e)) = α(di(e)) (i = 0, 1) and such that α(e) ∈ E(Λ), for every e ∈ E();in other words, α sends vertices to vertices and edges to edges and α preserves the graph structure. A group G acts on a graph if it acts on the set and di(gm) = gdi(m), for all g ∈ G, m ∈ and i = 0, 1 (note that then G acts on V()and on E()). If a group G acts on a graph , the quotient G\ is a graph in a natural way: V(G\) = G\V and di(Gm) = Gdi(m) (m ∈ ,i = 0, 1).

© Springer International Publishing AG 2017 429 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0 430 A Abstract Graphs

The Star of a Vertex

Let v ∈ V = V(). Deﬁne Star0(v) = e ∈ E() d0(e) = v and Star1(v) = e ∈ E() d1(e) = v .

Observe that Star0(v) and Star1(v) are subsets of , and they are not disjoint if and only if has a loop based at v. Deﬁne the ‘star’ of the vertex v to be

Star(v) = Star(v) = Star0(v) ∪. Star1(v)

The valency of v in is the cardinality of Star(v) (so, the valency of v is the number of edges e ∈ E() incident with v, i.e., having v as a vertex, where a loop is counted twice). Note that if α : −→ Λ is a morphism of abstract, then α(Stari(v)) ⊆ Stari(α(v)) (i = 0, 1). Therefore α induces a map of sets

αv : Star(v) −→ StarΛ α(v) , for each v ∈ V(). We say that α is an immersion if for each v ∈ V(), the induced map

αv : Star(v) −→ StarΛ α(v) is injective. Observe that if α fails to be an immersion, it is because there exists a pair of different edges {e1,e2} such that either d0(e1) = d0(e2) or d1(e1) = d1(e2) and α(e1) = α(e2). We say that α is a covering of abstract graphs if αv is a bijection, for all v ∈ V().

Paths

To make the notion of ‘path’ easier to describe, one introduces new formal symbols e±1 for each e ∈ E. For convenience, we make the identiﬁcation e1 = e,for −1 e ∈ E(). One extends the incidence maps di as follows: d0(e ) = d1(e) and −1 −1 −1 d1(e ) = d0(e); we also refer to the e as edges; one views e as the ‘inverse edge’ of e. Put E−1 ={e−1 | e ∈ E}; we assume that E−1 ∩ E =∅.If α : −→ Λ is a morphism of graphs, we extend α to the formal edges e−1 by setting α(e−1) = α(e)−1. Let ε = (ε1,...,εn), where εi =±1(i = 1,...,n) and n ≥ 1 is a natural number. Deﬁne Circn(ε) to be a graph with n vertices (that we take to be the elements of A.1 The Fundamental Group of an Abstract Graph 431

Z/nZ) and n edges e1,...,en

12e2

e1 e3

Circn(ε) : 0 3 .

en . . n − 1 such that d0(ei) = i − 1 and d1(ei) = i,ifεi = 1, and d0(ei) = i and d1(ei) = i − 1, if εi =−1. We refer to a graph of the form Circn(ε) as a circuit of length n (or n-circuit). Note that a circuit of length 1 is a loop. 0 • 0

A standard arc An(ε) of length n is a graph with n + 1 vertices 0, 1,...,n, and n edges e1,...,en 0 12n − 1 n An(ε) : •••··· •• e1 e2 en such that d0(ei) = i − 1 and d1(ei) = i,ifεi = 1, and d0(ei) = i and d1(ei) = i − 1, if εi =−1. We call 0 and n the initial and terminal vertices of the standard arc An(ε). Note that one obtains the circuit Circn(ε) by identifying the vertices 0 and n in An(ε). Deﬁne a path p = pv,w from a vertex v to a vertex w of to be a ﬁnite sequence ε1 εr ε1 = εr = εi = εi+1 e1 ,...,er of edges such that d0(e1 ) v, d1(er ) w and d1(ei ) d0(ei+1 ) for all i

f : Ar (ε) −→ (A.1) such that f(0) = v and f(r)= w. This way of deﬁning a path will be useful later in Sect. A.4 dealing with certain algorithms in free abstract groups. The underlying graph of the path pv,w is the subgraph of with edges e1,...,er (note that these may not be distinct) and their vertices. The path pv,w is called εi εi+1 ε −ε ∈ = reduced if it contains no subsequence ei ,ei+1 of the form e ,e (e E(), ε ±1). Note that if the path is deﬁned as a morphism from a standard arc, as in (A.1), then saying that it is reduced is the same as saying that f is an immersion. If v = w, then a path pv,v is called a cycle based at v or a closed path based at v (a cycle of length 0 is just a vertex; this is also called the ‘empty cycle’ based at v). 432 A Abstract Graphs

The underlying graph of a reduced cycle of length greater that 0 has a subgraph which is a circuit.

= ε1 εr = ε1 εr If p e1 ,...,er and p e1 ,...,er are paths in , we say that p and p are elementary homotopic if one can pass from p to p by deleting or inserting adjacent pairs of the form eε,e−ε (e ∈ E()); observe that if p and p are elementary homotopic, then they have the same initial and the same terminal vertices. This generates an equivalence relation on the set of paths, which is called homotopy;explicitly: two paths p and q in are homotopic if there exists a ﬁnite sequence of paths p = p1,...,pt = q such that any two adjacent paths pi,pi+1 are elementary homotopic. Note that every path is homotopic to a reduced path.

Let v be a fixed vertex of an abstract graph . Given two cycles pvv and pvv based at a vertex v, define their product pvvpvv by concatenation. This product is abs compatible with the homotopy relation defined above. Let π1 (, v) denote the set abs of homotopy classes of cycles based at v. Then it easily follows that π1 (, v) is a group whose identity element is the class represented by the empty cycle based at v: this is called the fundamental group of based at v. A more algebraic approach is the following. Put Y = ye e ∈ E() , = abs and let Φ Φ(Y) denote the free abstract group with basis Y . Then π1 (, v) is the subgroup of Φ consisting of those elements represented by words of the form ε1 ··· εr ε1 εr ye1 yer such that e1 ,...,er is a cycle in based at v; this interpretation of abs π1 (, v) is justified because homotopy in paths corresponds to the usual equiva- abs lence of words. In particular, one deduces that π1 (, v) is a free abstract group. One also obtains the following result from this interpretation and from the classical treatment of free groups (see Sect. B.2 in Appendix B).

abs Lemma A.1.1 An element of π1 (, v) is represented by a unique reduced cycle based at v, or equivalently, if p1 and p2 are reduced homotopic cycles based at v, then p1 = p2.

An abstract graph is connected if for any given pair of vertices v,w, there exists a path in from v to w. For an abstract graph deﬁne a relation R on as follows: for m, m ∈ , we say that mRm if there exists a path from d0(m) to d0(m ); this is an equivalence relation. The equivalence classes of R are connected subgraphs of , which we call the ‘connected components’ of . One easily checks that the image of a connected abstract graph under a morphism is connected. A connected abstract graph is a tree if it contains no circuits, or equivalently, if the only reduced cycles in have length 0.

Exercise A.1.2 Let be a abstract graph and let R be a ring (with an identity element 1). For a set X denote by [RX] the free R-module with basis X. Consider the sequence of free R-modules and homomorphisms

d ε 0 −→ RE() −→ RV () −→ R −→ 0, (A.2) A.1 The Fundamental Group of an Abstract Graph 433 where d(e) = d1(e) − d0(e) and ε(v) = 1 (e ∈ E(),v ∈ V ()). (a) Show that is connected if and only if the sequence (A.2) is exact at [RV ()]. (See Proposition 2.3.2.) (b) Show that is a tree if and only if the short exact sequence (A.2) is exact.

If v and w are vertices in a tree T , it is not hard to see that there is a unique reduced path in the tree from v to w; we denote by [v,w] the underlying graph of such a reduced path: it is a ﬁnite subtree of T , and we refer to [v,w] as the chain determined by v and w. abs If is connected, then the fundamental group π1 (, v) is independent of the chosen vertex v. This is a consequence of the following result, which also provides abs a basis for the free group π1 (, v). Using Zorn’s Lemma, one sees that every abstract connected graph contains a subtree T with V()= V(T) (a ‘maximal tree’).

Proposition A.1.3 Let be a connected abstract graph and let T be a maximal tree of . Choose v ∈ V(); let Y be as above and put X = ye e ∈ E() − E(T ) .

abs Then the restriction to π1 (, v) of the natural homomorphism of free abstract groups ϕ : Φ(Y) −→ Φ(X) which sends X to X identically and sends each y ∈ Y − X to 1, is an isomorphism abs from π1 (, v) onto Φ(X).

Proof Given e ∈ E(), deﬁne

− y = ρ y ρ 1 , e d0(e) e d1(e)

ε1 εr ε1 εr where, for a given vertex w ∈ V(T),wesetρw = ye ···ye ,ife ,...,er is 1 r 1 the unique reduced path in T from v to w. Note that ye represents an element of abs ∈ abs ∈ π1 (, v) (by abuse of notation we write ye π1 (, v)), and if e E(T ), then = ye 1. Define : −→ abs ≤ ψ Φ(X) π1 (, v) Φ(Y) = ∈ ∈ − to be the homomorphism that sends x ye X to ye (e E() E(T )). Then clearly ϕ| abs ψ = idΦ(X). π1 (,v) Next we need to show that ψϕ| abs = id abs . To prove this it is first π1 (,v) π1 (,v) ∈ = −1 convenient to extend our notation as follows: if e E(), we define ye−1 ye . Then we have

−1 −1 −1 − −1 −1 = = 1 = − − y ρd (e)yeρ ρd (e)y ρ ρ 1 y 1 ρ −1 . e 0 d1(e) 1 e d0(e) d0(e ) e d1(e ) 434 A Abstract Graphs

= −1 Hence, if we deﬁne y e−1 (ye) ,wehave

−1 y ε = ρ ε y ε ρ ε ,ε=±1,e∈ E(). e d0(e ) e d1(e )

ε1 εn abs = ··· = ε ··· εn ∈ Consider an element π ye1 yen y 1 ye π (, v) corresponding to a e1 n 1 ε1 εn cycle e1 ,...,en in based at v. Then we have

−1 = ε ··· εn = ··· ε = ··· π y 1 ye ρ ε1 y ε1 yeεn ρd1(e n ) y ε1 yeεn , e1 n d0(e ) e e

ε = ε = ε1 = = εn since ρd0(e 1 ) ρd1(e n ) 1, because d0(e ) v d1(e ). Thus

ψϕ| abs (π) = π, π1 (,v) = ∈ (since y εi 1, when ei E(T )), as desired. ei

The following corollary is now clear.

abs Corollary A.1.4 Let be a connected abstract graph. The group π1 (, v) is a free group which is independent, up to isomorphism, of the choice of the vertex v; furthermore, its rank is the cardinality of the set E() − E(T ), where T is a maximal tree of , and this is independent of the chosen maximal tree.

abs In view of this corollary, one sometimes uses the notation π1 (), rather than abs abs π1 (, v), when is a connected graph, and we refer to π1 () as the fundamental group of the abstract graph .

Corollary A.1.5 Let Λ be a connected subgraph of a connected graph . Choose ∈ ⊆ abs abs v V(Λ) V(). Then π1 (Λ, v) is naturally embedded into π1 (, v) as a abs abs free factor (i.e., a basis of the subgroup π1 (Λ, v) of π1 (, v) can be extended abs to a basis of π1 (, v)).

Proof Choose a maximal tree T of Λ. Extend T toamaximaltreeT of . Then Λ − T ⊆ − T . Continuing with the notation in the proof of Proposition A.1.3, { | ∈ − } abs we deduce that the basis ye e E(Λ) E(T ) of π1 (Λ, v) is a subset of the { | ∈ − } abs basis ye e E() E(T ) of π1 (, v).

The following algorithm is clear from the proof of Proposition A.1.3.

abs Algorithm A.1.6 Construction of a basis for π1 (, v) when is a connected ﬁnite graph.

Explicitly: We assume that is speciﬁed by its vertices, edges and explicitly given incident functions d0,d1. Since a maximal tree of is a subtree that covers all its vertices, one can construct algorithmically one such maximal tree T . For each A.2 Coverings of Abstract Graphs 435 edge e ∈ − T , construct a cycle e˜ based at v as the path deﬁned as follows: take the unique reduced path in T from v to d0(e), followed by e, followed by the unique ˜ abs path in T from d1(e) to v. Then e represents an element of π1 (, v), and the set abs of all those elements is a basis for π1 (, v).

Example A.1.7 Let G be an abstract group and let X be a subset of G with 1 ∈/ X. The Cayley graph Γ = Γ(G,X)of G with respect to X is deﬁned as follows: Γ = V(Γ)∪. E(Γ ), where V(Γ)= G, E(Γ ) = G × X, d0(g, x) = g and d1(g, x) = gx (g ∈ G, x ∈ X). Clearly Γ is connected if and only if G = X.

A.2 Coverings of Abstract Graphs

In this section we construct a ‘universal covering’ ˜ of a connected graph and show how any other covering of appears as an image of ˜ . We begin with some general properties of immersions and coverings.

Proposition A.2.1 Let ζ : −→ be an immersion of abstract connected graphs and let v ∈ V( ). (a) Let p and p be paths in with the same initial vertex v. If ζ(p)= ζ(p ), then p = p . (b) If p is a reduced path in , then ζ(p) is a reduced path in .

ε ε ={ 1 r } ={ ε1 εs } = Proof (a) Say p e1 ,...,er and p (e1) ,...,(es) . Since ζ(p) ζ(p ),

{ ε1 εr }={ ε1 εs } = = we have ζ(e1) ,...,ζ(er ) ζ(e1) ,...,ζ(es) . Hence r s, ζ(ei) ζ(ei) = = = ={ ε} = and εi εi , for all i 1,...,r. Assume ﬁrst that r 1, i.e., p e1 and p

{ ε1 } = = = ε = ε1 (e1) , , ζ(e1) ζ(e1) and ε1 ε1. Since also v d0(e1) d0((e1) ), we deduce = = that either d0(e1) d0(e1) or d1(e1) d1(e1). Hence, since ζ induces an injection

= ={ ε}={ ε1 }= on Star(v),wealsohavethate1 e1; therefore p e1 (e1) p , proving the result for paths of length 1. The general result now follows by an easy induction. (b) We think of the path p as a morphism from a standard arc, p : A(ε) → . Since p is reduced, the morphism p is an immersion. So the result reduces to the obvious statement that the composition of immersions is an immersion.

Proposition A.2.2 Let ζ : −→ be a covering of connected abstract graphs and let v ∈ V( ). (a) If p is a path in with initial vertex ζ(v ), then there exists a unique path p˜ (the ‘lifting’ of p) in with initial vertex v such that ζ(p)˜ = p.

(b) If p1 and p2 are homotopic paths in with initial vertex ζ(v ), then p˜1 and p˜2 are homotopic paths in .

(c) If ζ has a section (i.e., a graph morphism σ : −→ such that ζσ = id), then ζ is an isomorphism. 436 A Abstract Graphs

Proof Parts (a) and (b) are easy to prove. Part (c) follows from (a) and the connectedness of .

Proposition A.2.3 Let ζ : −→ be an immersion of connected abstract graphs and let v ∈ V( ). Then the natural homomorphism

: abs −→ abs ζ π1 ,v π1 ,ζ v is injective.

= ∈ abs Proof Let 1 α π1 ( ,v ); by Lemma A.1.1, α is represented by a unique cycle p based at v in reduced form. Since α = 1, |p|≥1. Hence |ζ(p)|≥1. By Proposition A.2.1(b), ζ(p) is in reduced form; therefore, ζ(α)= 1.

Associated with a connected abstract graph one can define an abstract ‘universal covering graph’ ν : ˜ −→ of as follows. The morphism ν is surjective and a covering of the abstract graph , and the following universal property is satisfied: given a surjective covering graph ζ : Γ −→ Γ of an abstract graph Γ , vertices v ∈ V()˜ , w ∈ V(Γ ) and a morphism of graphs ϕ : −→ Γ such that ϕν(v) = ζ(w ), then there exists a unique morphism of graphs ϕ˜ : ˜ −→ Γ such that ϕ(v)˜ = w . From this definition it follows easily that an abstract universal covering graph ν : ˜ −→ is unique up to isomorphism, if it exists. We proceed to its construction. ˜ abs To define first we choose a maximal tree T of , and identify π1 () with the free group Φ(X) as in Proposition A.1.3, where X = ye e ∈ E() − E(T ) .

Furthermore, we require the existence of a morphism of graphs ν : ˜ −→ and we abs ˜ want the fundamental group π1 () to act freely on in such a way that ν induces abs \ ˜ an isomorphism of the quotient graph π1 () and . This forces ˜ = abs × ˜ = abs × π1 () and V() π1 () V (). ˜ To complete the requirements we need to specify the incidence maps di : −→ V()˜ (i = 0, 1) on the edges of ˜ . Put

d0(g, e) = g,d0(e) ,

= ¯ ∈ abs ∈ d1(g, e) gye,d1(e) , g π1 (), e E() ,

¯ abs = ¯ ∈ where ye is the image of ye in π1 () Φ(X), i.e., ye is 1 if e E(T ), and it is ye otherwise. To facilitate the calculations later it is convenient to observe that the A.2 Coverings of Abstract Graphs 437 above deﬁnitions for d0 and d1 extend to the following formulas valid also for the inverse edges:

ε = ¯ε(−η(ε)) ε d0(g, e) gye ,d0 e ,

ε = ¯ε(1−η(ε)) ε d1(g, e) gye ,d1 e , ∈ abs ∈ =± = − = where g π1 (), e E(), ε 1, η(1) 0 and η( 1) 1. Deﬁne ν : ˜ −→ abs = ˜ to be the natural projection. The group π1 () Φ(X) acts freely on by multiplication on the ﬁrst component. Note that Λ ={(1,m)| m ∈ } is a connected subgraph of ˜ : it contains an isomorphic copy

T = (1,m) m ∈ T of T , and d0(1,m)∈ T for all (1,m)∈ Λ. We shall show in Proposition A.2.5 that ν : ˜ −→ , as constructed above, is indeed a universal covering graph of .

Proposition A.2.4 abs (a) For every subgroup H of π1 (), the induced morphism ˜ νH : H\ −→ is a covering of abstract graphs. (b) ˜ is a tree.

Proof (a) is clear. Let Λ be as deﬁned above. To prove (b) ﬁrst observe that if ˜ e ∈ X = E() − E(T ), then Λ ∪ yeΛ is a connected subgraph of because ∈ ∈ ∪ −1 d0(1,e) T and d1(1,e) yeT ; consequently, Λ ye Λ is also connected. It follows inductively that

Λ ∪ yε1 Λ ∪ yε1 yε2 Λ ∪···∪yε1 ···yεr Λ e1 e1 e2 e1 er ∈ =± ˜ = abs is connected, for ye1 ,...,yer X, εi 1. Since π1 ()Λ and since X gen- abs ˜ erates π1 (), one deduces that is connected (see Lemma 2.2.4(a)). To show that ˜ does not contain a circuit, we proceed by contradiction. Assume that there exists a nontrivial reduced cycle

ε1 εr (g1,e1) ,...,(gr ,er ) (A.3)

˜ ∈ ∈ abs =± εi = in of length r>0(ei E(),gi π1 (), εi 1). Since d1(gi,ei) ε + ε ε d0(gi+1,ei+1) i 1 (i = 1,...,r− 1) and d1(gr ,er ) r = d0(g1,e1) 1 , we deduce that

ε (1−η(ε )) εi εi+1(−η(εi+1)) εi+1 g y¯ i i ,d e = g + y¯ ,d e (i = ,...,r − ), i ei 1 i i 1 ei+1 0 i+1 1 1

− − g y¯εr (1 η(εr )),d eεr = g y¯ε1( η(ε1)),d eε1 . r er 1 r 1 e1 0 1 438 A Abstract Graphs

− = ¯εi ( η(εi )) Therefore, putting ai giyei , we obtain

ε a y¯ i = a + (i = ,...,r − ), i ei i 1 1 1 a y¯εr = a . r er 1

abs Hence, in π1 () we have y¯ε1 ···y ¯εr = 1. e1 er ε1 ··· εr ε1 εr Consider now the element ye1 yer of Φ(Y). Since e1 ,...,er is a closed path ε1 ··· εr = in , we deduce from Proposition A.1.3 that ye1 yer 1inΦ(Y). To obtain ε1 ··· εr the desired contradiction we shall show that ye1 yer is in reduced form. In- −ε + εi = i 1 = =− ei = deed, if we had yei yei+1 , then ei ei+1 and εi εi+1. Then d1(qi,ei) ε ( −η(ε )) (−ε )(−η(−ε )) ei+1 ¯ i 1 i = ¯ i i = d0(qi+1,ei+1) implies that qiyei qi+1yei . Hence qi ε ε + qi+1, and therefore the edges (qi,ei) i and (qi+1,ei+1) i 1 would be inverse to each other, contrary to our assumption that (A.3) is reduced.

Proposition A.2.5 Let be a connected abstract graph, and let ˜ be the graph constructed above. (a) ν : ˜ −→ is an abstract universal covering of . (b) Let ζ : Γ −→ Γ be a covering of abstract graphs, and let ϕ : −→ Γ be a morphism of graphs. Let w ∈ V(Γ ) and v ∈ V() be vertices such that = abs ≤ abs ϕ(v) ζ(w ). Assume that ϕ(π1 (, v)) π1 (Γ ,w ); then there exists a unique morphism ϕ : −→ Γ such that ζϕ = ϕ and ϕ (v) = w .

˜ abs ˜ = ∈ Proof Since is a tree, π1 () 1. So (a) is a special instance of (b). Let u V() and let pv,u be a path in from v to u. Since ζ is a covering, there exists a unique lifting of the path ϕ(pv,u) in Γ to a path p in Γ with initial vertex w (see Proposition A.2.2(a)). Define ϕ (u) to be the terminal vertex of p . Observe abs ≤ abs that ϕ (u) is well-defined because of the condition ϕ(π1 (, v)) π1 (Γ ,w ). Finally, if e is an edge in incident with u, define ϕ (e) to be the unique edge e in Γ incident with ϕ (w ) such that ζ(e ) = ϕ(e).

Proposition A.2.6 Let be a connected graph and let H be a subgroup of abs π1 (, v), where v is a ﬁxed, but arbitrary, vertex of . Then there exists a covering ζ : −→ abs = = abs such that ζ(π1 ( ,v )) H , where ζ(v ) v, and the index of H in π1 (, v) | −1 | abs =∼ is ζ (v) . In particular, π1 ( ,v ) H (see Proposition A.2.3). Furthermore, this covering is unique up to an isomorphism commuting with ζ . Explicitly,

= \ ˜ = \ abs × H H π1 (, v) . A.2 Coverings of Abstract Graphs 439

Proof The uniqueness follows from Proposition A.2.5(b). Denote by

: ˜ −→ \ ˜ = \ abs × νH H H π1 (, v) the natural epimorphism of graphs; observe that νH is a covering of abstract graphs, ∈ ∈ abs \ ˜ and so it respects path homotopy. For h H , define ρH (h) π1 (H , (H1,v)) to be the homotopy class of cycles based at (H 1,v) represented by the cycle ˜ νH (p(1,v),(h,v)), where p(1,v),(h,v) is a path from (1,v) to (h, v) in . Then ρH is well-defined since νH respects path homotopy. Furthermore, ρH is a homomor- ˜ phism. Note that, since is a tree, we can choose p(1,v),(h,v) in the above definition to be the unique reduced path from (1,v)to (h, v). ˜ Since νH is a covering, a reduced cycle in H\ based at (H 1,v) can be lifted uniquely to a reduced path in ˜ with initial vertex at (1,v)(which necessarily ends at a vertex of the form (h, v),forsomeh ∈ H ). Therefore, ρH is a bijection. It is \ abs × −→ now clear that the projection (H π1 (, v)) is the desired covering.

Example A.2.7 Let X be a set and consider the graph consisting of a single vertex v and a set of edges {ex | x ∈ X} in a one-to-one correspondence with X,so that di(ex) = v, where x ∈ X and i = 0, 1(i.e., is a bouquet of |X| loops). Then abs = the fundamental group π1 () of is the free group Φ Φ(X) with basis X. Moreover, the abstract universal covering graph ˜ coincides with the Cayley graph Γ(Φ,X)of the free group Φ with respect to X: its vertices are the elements of Φ (which we identify with Φ ×{v}) and its edges are the pairs (f, x) ∈ Φ × X with d0(f, x) = f and d1(f, x) = fx. As we have shown above, Γ(Φ,X)is a tree.

Using the notation in this example, Proposition A.2.6 translates into the following corollary.

Corollary A.2.8 Let Φ = Φ(X) be a free abstract group with basis the set X, and let Γ = Γ(Φ,X)be the Cayley graph of Φ with respect to X. (a) Given a subgroup H of Φ, consider the map

: −→ abs \ = abs \ ρH H π1 (H Γ) π1 (H Γ,H1) ∈ ∈ abs \ that sends h H to the element ρH (h) π1 (H Γ)determined by the image in H \Γ of the unique reduced path p1,h from 1 to h in Γ(Φ,X). Then ρH is an isomorphism. (b) If H ≤ K are subgroups of Φ, the following diagram commutes

abs \ abs \ π1 (H Γ) π1 (K Γ)

ρH ρK

H K 440 A Abstract Graphs

where the upper horizontal map is the homomorphism induced by the natural projection H \Γ −→ K\Γ .

It is worth noticing that the NielsenÐSchreier subgroup theorem, which asserts that a subgroup of an abstract free group is also free, is an immediate consequence of part (a) in the above corollary and Corollary A.1.4. We end this section with a result that shows how a given immersion can be ‘en- larged’ to a covering, in certain cases.

Lemma A.2.9 Let ζ : −→ be an immersion of connected graphs. Assume that has a unique vertex w, and that the set of vertices V( ) of is ﬁnite. Then there exists a covering

ζ¯ : ¯ −→ of abstract graphs such that (a) is a subgraph of ¯ with V( ) = V()¯ , and (b) ζ¯ extends ζ .

Proof Put V = V( ), E = E( ). Given an edge e ∈ E(), deﬁne Re = u ,v ∈ V × V ∃e ∈ E with d0 e = u ,d1 e = v ,ζ e = e .

Note that Re may be the empty set. Since ζ is an immersion, Re defines an injective map from a subset of V to a subset of V . Since V is finite, we may choose a subset ¯ Se of V × V containing Re such that Se defines a bijection V −→ V . Define as follows: V¯ = V()¯ = V( ) = V . The set of edges E¯ = E()¯ of ¯ is ¯ E = u ,v ,e e ∈ E(), u ,v ∈ Se .

The incidence maps of ¯ are

¯ d0 u ,v ,e = u ,d1 u ,v ,e = v u ,v ,e ∈ E .

Next deﬁne ζ¯ : ¯ −→ by

ζ¯ w = w, ζ¯ u ,v ,e = e w ∈ V¯ = V , u ,v ,e ∈ E¯ , where w is the unique vertex of . Then ζ¯ is a morphism of graphs, and the construction shows that it is a covering. ¯ Finally, deﬁne ι : −→ by ι(v ) = v , ι(e ) = (d0(e ), d1(e ), ζ(e )) (v ∈ V ,e ∈ E ). Clearly ι is an embedding of graphs, and ζ = ζι¯ . A.3 Foldings 441

A.3 Foldings

Let α : −→ Λ be a morphism of graphs. As was pointed out in Sect. A.1,ifα is not an immersion it is because there exists a pair of different edges {e1,e2} in such that either d0(e1) = d0(e2) or d1(e1) = d1(e2) and α(e1) = α(e2). Motivated by this observation, we make the following construction: given two edges e1,e2 ∈ E() such that either d0(e1) = d0(e2) or d1(e1) = d1(e2), we deﬁne a new graph

/[e1 = e2] obtained from by identifying e1 with e2, d0(e1) with d0(e2) and d1(e1) with d1(e2). Following a terminology due to Stallings, we call this construction a folding. The natural quotient map = : −→ [ = ] f fe1,e2 / e1 e2 is an epimorphism of graphs. The following proposition is clear.

Proposition A.3.1 Let α : −→ Λ be a morphism of graphs.

(a) Let e1,e2 ∈ E() be edges such that either d0(e1) = d0(e2) or d1(e1) = d1(e2) = and α(e1) α(e2). Then α factors through the folding fe1,e2 , i.e., = α α fe1,e2 ,

for some unique morphism α : /[e1 = e2]−→Λ. (b) Assume that is ﬁnite. Then α factors through a ﬁnite sequence of foldings and an immersion, i.e., there is a graph , a composition of foldings

f1 f2 fr−1 fr = 1 −→ 2 −→···−→ r −→ 

and an immersion ι : −→ Λ such that

α = ιf r ···f1.

The sequence of foldings in part (b) of the proposition above is in general not unique, but one shows easily by induction on the number of edges of that and ι are uniquely determined by α. Moreover, and ι are obtained algorithmically: at each stage i one simply has to check in the ﬁnite graph i whether there is a pair e1,e2 ∈ E() with either d0(e1) = d0(e2) or d1(e1) = d1(e2) and α(e1) = α(e2).

Proposition A.3.2 Let be a connected graph and let e1 and e2 be edges of with a common vertex v such that either d0(e1) = d0(e2) or d1(e1) = d1(e2). Let ¯ f : −→ = /[e1 = e2] 442 A Abstract Graphs be the corresponding folding map. Then f induces an epimorphism on the corre- abs = abs ¯ sponding fundamental groups, i.e., f(π1 (, v)) π1 (, f (v)).

= abs ≤ abs ¯ Proof Deﬁne H f(π1 (, v)); then H π1 (, f (v)). By Proposition A.2.6 : −→ ¯ abs = ∈ there exists a covering ζ with ζ(π1 ( ,v )) H , where v V(). By Proposition A.2.5(b) there exists a unique morphism f : −→ such that f (v) = v and ζf = f . Therefore, since ζ is a covering, we have f (e1) = f (e2); hence by Proposition A.3.1(a), there exists a morphism ζ : ¯ −→ such that f = = = = ζ f . Hence ζζ f ζf f .Soζζ id¯ . It follows from Proposition A.2.2(c) = abs ¯ that ζ is an isomorphism of graphs, and thus H π1 (, f (v)), as needed.

A.4 Algorithms

In this section we describe several algorithms related to finitely generated subgroups of a free group Φ of finite rank n with basis X. We think of Φ as the fundamental abs group π1 (, u), where is a finite graph with a single vertex u and edges labeled by X: a bouquet of n loops (see Example A.2.7). An element of Φ is represented byacyclein based at u, and a subgroup of is represented by a finite set of generators of that subgroup.

Algorithm A.4.1 Represents a ﬁnitely generated subgroup H of Φ as an immersion ι : Γ → with Γ a ﬁnite connected graph.

{ } abs = We assume that H is generated by a ﬁnite subset h1,...,hn of π1 (, u) Φ and for each i we are given a cycle pi in based at u representing hi . We think of pi as a morphism of graphs

pi : Bi −→ , where Bi is a standard arc (see (A.1)). Construct a graph B to be the disjoint union of the graphs B1,...,Bn, and deﬁne a new graph Γ1 obtained from B by identifying all the initial and terminal vertices of all the Bi to a vertex that we will denote by v. Denote the image of Bi in Γ1 by Ci ; this is a circuit. Hence Γ1 is a bouquet of the n circuits C1,...,Cn joined abs at the vertex v. Therefore π1 (B, v) is a free group of rank n with a basis whose elements are represented by the closed paths deﬁned by the image of each of those circuits. Since, for each i, pi maps the initial and terminal vertices of Bi to u,themor- phisms p1,...,pn induce a morphism of graphs

α : Γ1 −→ that sends v to u. It follows that

abs = α π1 (B, v) H. A.4 Algorithms 443

Next apply Proposition A.3.1 to obtain a sequence of foldings f1,...,fr and an immersion ι

f1 fr−1 fr ι Γ1 −→···−→ Γr −→ Γ −→ such that α = ιf r ···f1. By Proposition A.3.2, a folding induces a surjection of ··· = abs = fundamental groups. So, if we put fr f1(v) w,wehaveιπ1 (Γ, w) H .Soι is the desired immersion.

Algorithm A.4.2 Obtains a basis of a subgroup H of Φ when H is given by a ﬁnite set of generators.

We continue with the above set-up. Since ι is an immersion, we deduce from Proposition A.2.3 that the induced map

˜ : abs −→ ι π1 (Γ, w) H is an isomorphism of groups. As pointed out above, the graph Γ and the immersion ι can be obtained algorithmically. So it sufﬁces to describe how to obtain a basis for abs π1 (Γ, w) algorithmically, and this is done in Algorithm A.1.6.

Algorithm A.4.3 Decides whether an element of a free group is in a ﬁnitely generated subgroup.

We are given n cycles p1,...,pn in based at u. Each pi represents an element abs = = hi of π1 (, u) Φ(i 1,...,n).LetH be the subgroup of Φ generated by h1,...,hn−1. We describe next an algorithm to decide whether hn ∈ H or not. We continue with the notation in Algorithm A.4.1.Let

= ∪···∪ Γ1 C1 Cn−1.

Denote by Γ and Γ the images in Γ of Γ1 and Cn, respectively. Let ι be the restriction of ι to Γ . Since ι is an immersion, so is ι , and hence the immersion : → ∈ abs = ι Γ represents H . It follows that hn H if and only if π1 (Γ ,w) abs abs π1 (Γ, w), because ι is injective on π1 (Γ, w). And this can be decided by checking whether Γ determines a cycle in Γ .

Intersection of Finitely Generated Subgroups

Next we want to prove that the intersection of two finitely generated subgroups of a free group is finitely generated, and that a basis for it can be obtained algorithmically. Obviously one may assume that the free group has finite rank. 444 A Abstract Graphs

Theorem A.4.4 Let H1 and H2 be ﬁnitely generated subgroups of a free abstract group Φ of ﬁnite rank. Then

(a) the subgroup H1 ∩ H2 of Φ has ﬁnite rank, and (b) there exists an algorithm to construct a basis for H1 ∩ H2, assuming that H1 and H2 are given by explicit ﬁnite sets of generators.

= abs Proof We assume as above that Φ π1 (, u), where is finite graph with a single vertex u and n loops. Using Algorithm A.4.1, we assume that Hi is represented by an immersion ιi : Γi → , where Γi is a finite connected graph (i = 1, 2). The first step is to show how to represent H1 ∩ H2 by an immersion. To do this we begin by recalling what is the pullback diagram of ι1 and ι2

ϕ1 Γ Γ1

ϕ2 ι1

Γ2 ι2

Here Γ is a graph deﬁned as follows: Γ = (m1,m2) ∈ Γ1 × Γ2 ι1(m1) = ι2(m2) ; its set of vertices is V(Γ)= (v1,v2) ∈ V(Γ1) × V(Γ2) ι1(v1) = ι2(v2) , and the incidence maps di : Γ → V(Γ)are deﬁned as

d0(m1,m2) = (d0m1,d0m2), d1(m1,m2) = (d1m1,d1m2).

Finally, the morphisms ϕ1 and ϕ2 are just the projection maps. It is clear that this diagram satisﬁes the universal property of a pullback: if ψi : Σ → Γi (i = 1, 2) are maps of graphs with ι1ψ1 = ι2ψ2, then there is a unique map of graphs ψ : Σ → Γ with ϕ1ψ = ψ1 and ϕ2ψ = ψ2. This deﬁnition provides an algorithm to construct Γ from Γ1 and Γ2. In general Γ is not a connected graph. Put ι = ι1ϕ1 = ι2ϕ2. We claim that ι : Γ →

is an immersion. Indeed, let (m1,m2) and (m1,m2) be edges in Γ with the same ini- = tial vertex (respectively, the same terminal point) (v1,v2).If(m1,m2) (m1,m2), = = then either m1 m1 or m2 m2; since ι1 and ι2 are immersions, it follows that = ι(m1,m2) ι(m1,m2). On the other hand, if (m1,m2) is a loop based at (v1,v2), then both m1 and m2 are loops. It follows that ι induces an injection on Star(v1,v2), proving the claim. A.5 Notes, Comments and Further Reading 445

abs = abs = ∈ = By assumption ι1(π1 (Γ1,v1)) H1, ι2(π1 (Γ2,v2)) H2, where vi Γi (i 1, 2), ι1(v1) = ι1(v2) = u, and u is the unique vertex of .Letv = (v1,v2) ∈ Γ . abs ⊆ ∩ Then we clearly have ι(π1 (Γ, v)) H1 H2. abs = ∩ ∈ In fact, we assert that ι(π1 (Γ, v)) H1 H2. To see this assume that h ∩ ≤ = abs ∈ abs = = H1 H2 Φ π1 (, u).Lethi π1 (Γi,vi) be such that ιi(hi) h(i 1, 2). Represent hi by a reduced cycle pi based at vi ; since ιi is an immersion, ιipi is a reduced cycle of based at u(i= 1, 2) (see Proposition A.2.1(b)) representing h, i.e., the cycles ι1p1 and ι2p2 are homotopic and reduced. We deduce from Lemma A.1.1 that ι1p1 = ι2p2. Interpreting paths as morphisms from standard arcs (see (A.1)), this is equivalent to saying that there exist a standard arc Ar (%) and morphisms

p1 : Ar (%) → Γ1 and p2 : Ar (%) → Γ2 such that the compositions ι1p1 and ι2p2 are equal. Using the pullback property, we deduce that there exists a morphism p : Ar (%) → Γ with ϕ1p = p1 and ϕ2p = p2. Hence p is a cycle in Γ based at v = (v1,v2), i.e., it represents an element k of abs = π1 (Γ, v); moreover, ιk h. This proves the assertion. Next consider the connected component Γ3 of Γ containing v = (v1,v2), and denote by ι3 : Γ3 → the restriction of ι to Γ3. Then ι3 is an immersion of ﬁnite connected graphs and so ι3 represents H1 ∩ H2, i.e., ι3 induces an explicit isomor- abs → ∩ phism π1 (Γ3,v) H1 H2. Therefore, using Algorithm A.1.6, one obtains a basis for H1 ∩ H2. This proves both parts (a) and (b).

A.5 Notes, Comments and Further Reading

There is a very extensive mathematical literature on abstract graphs, often related to combinatorics. The point of view that we adopt here regarding abstract graphs is very much determined by the topics in this book. For more detailed treatments of abstract graphs and actions of groups on them one can consult Serre (1980), Dicks and Dunwoody (1989) or Stallings (1983). The ideas for the algorithms in Sect. A.4 are due to Stallings. Part (a) of Theorem A.4.4 is due to Howson (1954). The concept of abstract graph that we use in this Appendix is substantially the same as that used in the BassÐSerre theory of groups acting on trees developed in Serre (1980). In the latter book a graph includes by deﬁnition both the set of edges E() and the set of inverse edges E()−1; then what we denote by E() in our setting corresponds to a speciﬁc ‘orientation’ in the set-up of Serre. Our notion of action of a group on a graph corresponds to what Serre calls ‘action without inversion’, and our notion of morphism or map of graphs would correspond, in Serre’s set-up, to morphisms that preserve given orientations. The standard constructions of abstract graphs arising from ‘free constructions’ of abstract groups, such as the Cayley graph of a group, the tree canonically associated to a free product of abstract groups or an amalgamated product of abstract groups, or, more generally, the universal covering graph of a graph of abstract groups, all come equipped with a natural orientation (i.e., a graph in the sense used in this book). Appendix B Rational Sets in Free Groups and Automata

In this Appendix it is shown that a rational subset of an abstract free group can be described as the recognizable language of an automaton over an appropriately chosen alphabet.

Notation If Y is a set, Y ∗ denotes the free monoid on Y . The elements w = ∗ y1 ···ym of Y are called words on the alphabet Y(y1,...,yn ∈ Y).Thelength ∗ of w = y1 ···ym is m.Alanguage on the alphabet A is simply a subset of A .

B.1 Finite State Automata: Review and Notation

A ﬁnite state automaton is a 4-tuple,

A = (A,Q,i,T) where Q and A are ﬁnite sets, i ∈ Q, T ⊆ Q, together with a function (the next state function) Q × A −→ P(Q) (B.1) that we denote by (q, a) → qa (q ∈ Q, a ∈ A) [P(Q) denotes the set of subsets of Q]. The set A is called the alphabet of the automaton A; Q is the set of states of A; i is the initial state of A; and T is the subset of terminal states of A. If Q ⊆ Q and a ∈ A, then Q a is deﬁned to be the union Q a = q a. q ∈Q

Then one extends the function (B.1)toafunction

Q × A∗ −→ P(Q)

© Springer International Publishing AG 2017 447 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0 448 B Rational Sets in Free Groups and Automata by the formulas

∗ q1 = q and q(wa) = (qw)a w ∈ A ,a∈ A .

The language recognized by the automaton A is the subset of A∗ L(A) = w ∈ A∗ iw ∩ T = ∅ .

A subset L of A∗ is termed recognizable if there exists some ﬁnite state automaton A with alphabet A such that L = L(A). A fundamental result, due to Kleene, says that the recognizable subsets of A∗ are precisely the rational subsets Rat(A∗) of A∗ (see Theorem 12.3.2).

Remark The definition that we have given for a finite state automaton corresponds to what is usually called a ‘nondeterministic’ and ‘complete’ finite state automaton: nondeterministic because the values qa of the next state function (B.1) are subsets of Q, rather than elements of Q, i.e., singleton subsets of Q; and complete because that function is defined on the whole set Q × A, rather than on just a subset. Actually, this is not an essential distinction in the sense that a subset L of A∗ is recognized by a nondeterministic and complete finite state automaton if and only it is recognized by some deterministic and not necessarily complete finite state automaton (see, for example, Eilenberg 1974).

B.2 The Classical Function ρ

Let X ={x1,...,xn} be a finite set of size n. Define Z to be the finite set with 2n letters

Z ={x1,...,xnx¯1,...,x¯n} (B.2) where x →x ¯ is an involution on Z, i.e., x¯ = x,forx ∈ Z. Two words in Z∗ are equivalent if one can pass from one to the other by a ﬁnite sequence of insertions or deletions of subwords of the form xx(x¯ ∈ Z).Aword w ∈ Z∗ is reduced if it does not contain subwords of the form xx(x¯ ∈ Z). Deﬁne

ρ : Z∗ −→ Z∗ as follows: if w ∈ Z∗, ρ(w) is the reduced word obtained by deleting, from left to right, all pairs xx(x¯ ∈ Z). Properties of ρ ∗ (ρ1)ρ(w1w2) = ρ(ρ(w1)w2) (w1,w2 ∈ Z ). (ρ2)ρρ(w)= ρ(w) (w ∈ Z∗). (ρ3)ρ(w)= w,ifw ∈ Z∗ is reduced. ∗ (ρ4)ρ(w1xxw¯ 2) = ρ(w1w2) (w1,w2 ∈ Z , x ∈ Z). ∗ (ρ5)ρ(w1w2) = ρ(ρ(w1)ρ(w2)) (w1,w2 ∈ Z ). B.3 Rational Subsets in Free Groups 449

Properties (ρ1), (ρ2) and (ρ3) follow from the deﬁnition. Property (ρ4) follows from (ρ1). Property (ρ5) is proved by induction on the length of w2. If two words w1 and w2 are reduced and equivalent, one proves, using (ρ4), that ρ(w1) = ρ(w2); and by (ρ3), w1 = w2. Therefore an equivalence class of words contains exactly one word which is reduced.

Proposition B.2.1 The subset ρ(Z∗) of Z∗ is recognizable.

Proof Note that ρ(Z∗) = Z∗ − Z∗{xx,¯ xx¯ | x ∈ X}Z∗. Since both Z∗ and {xx,¯ xx¯ | x ∈ X} are recognizable, so is ρ(Z∗) (see Theorem 12.3.2(c)).

B.3 Rational Subsets in Free Groups

Let B be a ﬁnite set and let Y be a subset of B∗. Deﬁne a function

∗ ∗ λY : B −→ P B

∗ ∗ ∗ from B into the set of subsets of B as follows: if w,w ∈ B , then w = b1 ···br ∈ λY (w) (bi ∈ B)ifw has the form

w = y0b1y1 ···yr−1br yr , where y0,...,yr ∈ Y and r ≥ 0. ∗ For example, if Y ={1}, then λY (w) ={w}, for all w ∈ B ; while if B = −1 Z ={x1,...,xn, x¯1,...,x¯n} (as in Sect. B.2) and Y = ρ (1), then the words in λY (w) are those obtained from w by deleting subwords of the form xx¯ or xx¯ (x ∈{x1,...,xn}).

∗ Lemma B.3.1 Let B be a ﬁnite set and let Y ⊆ B . Then the function λY sends recognizable subsets of B∗ to recognizable subsets of B∗.

Proof Let K be a recognizable subset of B∗. Say K = L(B), where B is a deterministic ﬁnite state automaton:

B = (B,Q,i,T).

For p,q ∈ Q, deﬁne a subset of B∗ ∗ Kp,q = w ∈ B pw = q .

To prove the result we shall construct a ﬁnite state automaton

B = B,Q ,i ,T 450 B Rational Sets in Free Groups and Automata

on the same alphabet B that recognizes λY (K).LetQ = Q ∪. {i }, where i is a new element not in Q;wetakei as the initial state of B . The subset T of terminal states of B is T, if Y ∩ K =∅; T = T ∪{i }, if Y ∩ K = ∅.

To complete the description of the automaton B we need a next state function

Q × B −→ P Q , which is deﬁned by

q ∈ pb if and only if bY ∩ Kp,q = ∅ (b ∈ B,p,q ∈ Q);

q ∈ i b if and only if YbY ∩ Kp,q = ∅ (b ∈ B,q ∈ Q).

Now one easily checks that λY (K) = L(B ).

We continue with the notation in Sect. B.2:

X ={x1,...,xn}⊆Z ={x1,...,xn, x¯1,...,x¯n}.

Proposition B.3.2 The function ρ : Z∗ −→ Z∗ sends recognizable subsets to recognizable subsets: if K is a recognizable subset of Z∗, then so is ρ(K).

Proof Deﬁne Y = ρ−1(1).Letw ∈ Z∗; then, as pointed out above,

λY (w) is the subset of Z∗ consisting of those words obtained from w by deleting subwords of the form xx¯ or xx¯ (x ∈{x1,...,xn}). Hence, ρ(w) is the unique element in ∗ ∗ λY (w) ∩ ρ(Z ). Therefore, if K is a recognizable subset of Z , then

∗ ρ(K)= λY (K) ∩ ρ Z is recognizable, because it is the intersection of recognizable subsets of Z∗ (see Theorem 12.3.2(c), Lemma B.3.1 and Proposition B.2.1).

Let Φ = Φ(X) be the free abstract group on X. We think of an element of Φ as an equivalence class w˜ of a word w ∈ Z∗ under the equivalence relation defined in Sect. B.2. With abuse of notation, one often denotes an element w˜ of Φ by w. Define a map ι : Φ −→ Z∗ by ι(w)˜ = ρ(w); this is well-defined because the equivalence class w˜ contains a unique reduced word. Then ι is an injection. B.4 Notes, Comments and Further Reading 451

Consider the commutative diagram

ι Φ Z∗

ρ ψ Z∗ where ψ is the unique morphism of monoids such that ψ(x)= x and ψ(x)¯ = x−1, for x ∈ X.

Theorem B.3.3 Let R be a rational subset of the free group Φ = Φ(X). Then ι(R) is a recognizable subset of the free monoid Z∗.

Proof By Lemma 12.3.1, there exists some R ∈ Rat(Z∗) with ψ(R ) = R.By Kleene’s theorem (Theorem 12.3.2), R is a recognizable subset of Z∗. Thus ι(R) = ιψ(R ) = ρ(R ) is recognizable, according to Proposition B.3.2.

B.4 Notes, Comments and Further Reading

There are many good general treatments of automata theory, such as Eilenberg (1974). Theorem B.3.3 is due to Benois (1969); the proof that we present here follows the treatment of Berstel (1979), Part III, Sect. 2, who in turn uses ideas from Fliess (1971). I am grateful to Benjamin Steinberg for bringing to my atten- tion Benois’ paper and to Jean-Eric Pin for very precise information about several proofs of Benois’ theorem. A different approach to Benois’ theorem can be found in Gilman (1987) and Steinberg (2001a), Theorem 26. Theorem B.3.3 provides an alternative to Algorithm A.4.3 to decide whether or not an element g of a free abstract group Φ = Φ(X), with finite basis X,isinagiven finitely generated subgroup of Φ. More generally, and this is what is in fact used in the proof of Theorem 12.3.10,ifH1,...,Hn are finitely generated subgroups of Φ (each of them given by a set of generators, i.e., words in X ∪ X−1), there is an explicitly constructed finite state automaton A on the alphabet Z (see (B.2)) such that ι(H1 ···Hn) = L(A). Hence, one can decide whether or not a given element g ∈ Φ is in the subset H1 ···Hn by checking whether or not ι(g) is recognized by A. The construction of A follows from the proof of Theorem B.3.3 and standard facts in automata theory: (1) given a finite subset Y of Z∗ one can explicitly describe a finite state automaton over the alphabet Z that recognizes Y ∗, and (2) given finitely many ∗ finite state automata A1,...,An over the alphabet Z , one can construct explicitly a finite state automaton A over the alphabet Z such that L(A) = L(A1) ···L(An). References

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Tang, C.Y.: Conjugacy separability of generalized free products of surface groups. J. Pure Appl. Algebra 120, 187Ð194 (1997) Tits, J.: Sur le groupe des automorphismes d’un arbre. In: Essays on topology and related topics (Mémoires dédiés à Georges de Rham), pp. 188Ð211. Springer, New York (1970) Tits, J.: A “theorem of LieÐKolchin” for trees. In: Contributions to algebra (collection of papers dedicated to Ellis Kolchin), pp. 377Ð388. Academic Press, New York (1977) Toinet, E.: Conjugacy p-separability of right-angled Artin groups and applications. Groups Geom. Dyn. 7(3), 751Ð790 (2013) Ventura, E.: Fixed subgroups in free groups: a survey. In: Combinatorial and Geometric Group Theory, New York, 2000/Hoboken, NJ, 2001. Contemp. Math., vol. 296, pp. 231Ð255. Amer. Math. Soc., Providence (2002) Weigel, T.: p-Projective groups and pro-p trees. In: Ischia group theory 2008, pp. 265Ð296. World Sci. Publ., Hackensack (2009) Weil, P.: Computing closures of finitely generated subgroups of the free group. In: Algorith- mic Problems in Groups and Semigroups, Lincoln, NE, 1998. Trends Math., pp. 289Ð307. Birkhäuser, Boston (2000) Wilson, J.S.: Profinite Groups. Clarendon Press, Oxford (1998) Wilson, J.S., Zalesskii, P.A.: Conjugacy separability of certain Bianchi groups and HNN extensions. Math. Proc. Camb. Philos. Soc. 123, 227Ð242 (1998) Wilton, H.: Hall’s Theorem for limit groups. Geom. Funct. Anal. 18, 271Ð303 (2008) Wise, D.T.: Subgroup separability of graphs of free groups with cyclic edge groups. Q. J. Math. 51(1), 107Ð129 (2000) Wise, D.T.: Subgroup separability of the Fig. 8 knot group. Topology 45(3), 421Ð463 (2006) You, S.: The product separability of the generalized free product of cyclic groups. J. Lond. Math. Soc. 56, 91Ð103 (1996) Zalesskii, P.A.: Geometric characterization of free constructions of profinite groups. Sib. Mat. Zh. 30(2), 73Ð84 (1989), 226 (Russian). Translation in: Siberian Math. J. 30(2), 227Ð235 (1989) Zalesskii, P.A.: Profinite groups, without free nonabelian pro-p subgroups, that act on trees. Mat. Sb. 181(1), 57Ð67 (1990) (Russian). Translation in: Math. USSR-Sb. 69(1), 57Ð67 (1991) Zalesskii, P.A.: Open subgroups of free profinite products. In: Proceedings of the International Conference on Algebra, Part 1, Novosibirsk, 1989. Contemp. Math., vol. 131, pp. 473Ð491. Amer. Math. Soc., Providence (1992) Zalesskii, P.A.: Normal divisors of free constructions of pro-finite groups, and the congruence kernel in the case of a positive characteristic. Izv. Ross. Akad. Nauk, Ser. Mat. 59(3), 59Ð76 (1995) (Russian). Translation in: Izv. Math. 59(3), 499Ð516 Zalesskii, P.A., Mel’nikov, O.V.: Subgroups of profinite groups acting on trees. Mat. Sb. 135(177)(4), 419Ð439 (1988), 559 (Russian). Translation in: Math. USSR-Sb. 63(2), 405Ð424 (1989) Zalesskii, P.A., Melnikov, O.V.: Fundamental groups of graphs of profinite groups. Algebra Anal. 1, 117Ð135 (1989). English translation: Leningrad Math. J. 1, 921Ð940 (1990) Zhou, W., Kim, G.: Subgroup separability of certain generalized free products of nilpotent-by-finite groups. Acta Math. Sinica, English Series, 1199Ð1204 (2013) Zieschang, H., Vogt, E., Coldewey, H-D.: Surfaces and Planar Discontinuous Groups. Lect. Notes Math., vol. 835. Springer, Berlin (1980) Index of Symbols

BG, 24 G = (G,π,T)—sheaf of pro-C groups, 137 B(X,∗)—bouquet of loops associated with G(S)—subgroup generated by the fibers Gs , (X, ∗), 63 s ∈ S, in a free product, 151 C—pseudovariety of finite groups, 5 GV —vertex subsheaf of G, 177 Cn(G, A)—homogeneous n-cochains, 21 Γ(G,X)—the Cayley graph, 39 C(Γ,R)—chain complex of a graph Γ , 46 Γ ∗(m)—connected profinite component, 38 ˜ CK (H )—centralizer, 1 Γ —the universal Galois C-covering graph of a CG(T )—kernel of action of G on T , 119 profinite graph Γ , 74 ¯ Circn(ε)—circuit, 431 G—induced graph of pro-C groups, 211 Circn(ε)—circuit of length n, 41 G(e), 178 ClC(H )—pro-C closure, 358 #G—order of G, 8 ClNil(H )—pronilpotent closure, 357 Gm—the stabilizer of an element m, 42 Cl(X)—closure in the pro-C topology, 329 Gm = gm | g ∈ G, 344 Clm(S)—closure in the topology of a free (G,π,Γ)—graph of pro-C groups, 177 monoid, 363 G(v), 178 Cn—cyclic group of order n, written G(ζ )—the group associated with a Galois multiplicatively, 2 covering ζ , 63 G CoindH (A)—coinduced module, 25 HG—core of the subgroup H in G, 281 = H H n(G, A)—cohomology group, 20 Cor CorG —corestriction map, 24 CorF —corestriction induced by a family of Hn(G, B)—homology group, 24 G ≤ subgroups, 262 H G, 1 ≤ C —multiplicative group of order p, 296 H c G, 1 p ≤ DMod(Λ)—category of discrete H o G, 1 Λ-modules, 12 H G, 1 Der(G, A)—group of derivations, 22 (( I G )) —augmentation ideal, 22 E(Γ )—the edge set of a graph Γ , 29 Ider(G, A)—group of inner derivations, 22 ∗ G E (Γ ), 46 IndH (B)—induced module, 25 = G/K Fix(ψ)—fixed subgroup of ψ, 317 Inf InfG —inflation map, 23 Fp—prime field with p elements or its KX(A)—constant sheaf over a space, 168 additive group, 2 KX(A)—free product of a constant sheaf over C G1 G2 = G1 G2—free pro-C a space, 168 product, 141 K(X,∗)(A)—free product of a pseudoconstant G2, 137 sheaf, 168 G/F, 147 K(M)—kernel of the monoid M, 365 G(Γ | )—the group associated with a Galois Kp(M)—p-kernel of the monoid M, 365 covering, 63 [[ Λ(X, ∗)]] —free profinite Λ-module, 12 G\Γ -quotient graph, 42 Lb—Tits line corresponding to b, 239

© Springer International Publishing AG 2017 461 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0 462 Index of Symbols

M = (M,π,T)—sheaf of profinite V(Γ)—vertex set of a graph Γ , 29 Λ-modules, 257 V−1W—Mal’cev product, 366 N—set of natural numbers, 1 [W]—smallest π-subtree containing W , 56 NK (H )—normalizer, 1 X , 400 PMod(Λ)—category of profinite and abelian groups, 400 Λ-modules, 12 and free groups, 400 C G C Π1 ( ,Γ)—fundamental pro- group, 180 and free-by-finite groups, 400 G Π (G,Γ)—fundamental group of a graph of and polycyclic-by-finite groups, 400 1 ∗ groups, 192 X —submonoid generated by X, 359 A ¯ R —tensor A-completion of a group, 419 X—closure in RCˆ, 329 RCˆ—pro-C completion, 8 X , 44 RE —the R-stabilizer of E pointwise, 403 X—closed subgroup generated by X, 6 RC(G)—kernel of the maximal pro-C quotient Z—group or ring of integers, 1 of G, 164 ZCˆ, 48 [RG]—abstract group algebra, 15 !Zπˆ , 51 [[ ]] C RG —complete group algebra, 15, 45 #t∈T Gt —internal# free product, 149 [ ] = G RX —free abstract R-module, 12 t∈T At T —direct sum indexed by the [[ RX]] —free profinite R-module on a profinite topological space T , 167 space X, 45 cd!p(G)—cohomological p-dimension, 26 G Rat(M)—family of rational subsets of M, 359 !T —external free product, 140 = G r C Res ResH —restriction map, 23 s∈S Gs —restricted free pro- product, 144 Rˆ—profinite completion, 8 d(G)—minimal number of generators of G, 6 dR —map in the chain complex C(Γ,R), 46 Rpˆ —pro-p completion, 8 S(G,Γ)—standard graph of a graph of d0(e)—initial vertex of an edge e, 29 groups, 195 d1(e)—terminal vertex of an edge e, 29 SC(G,Γ)—standard graph of a graph of μG, 137 C groups, 195 np( ), 48 C C S (G,Γ,H)—standard graph of a graph of π( )—set of primes dividing the order of C groups with respect to H , 195 some group in , 5 πabs()—fundamental group of , 434 S ⊆c T , 1 1 abs S ⊆o T , 1 π1 (, v)—fundamental group of at v, 432 C C Subgp(G)—space of closed subgroups π1 (Γ )—the fundamental pro- group of a of G, 145 profinite graph Γ , 75 T , 295 ρ—the classical function ρ, 448 G T —fixed points of T under the action vcdp(G), 290 of G, 113 [v,w]—chain connecting v and w, 56 ΥC(Γ, T )—(pro-C) universal Galois w(X), 154 C-covering of Γ , 82 w(X)—weight of a topological space X, 4 − Υ abs(Γ, T )—(abstract) universal covering x y = xyx 1, 381 − of Γ , 88 xy = y 1xy, 2 Index of Authors

A E Allenby, 424, 426 Engler, 324 Almeida, 217, 425 Evans, 426 Anissimow, 361 Aschenbrenner, 425 F Ash, 425 Ferov, 427 Auinger, 423 Fine, 422 Auslander, 346 Fliess, 451 Formanek, 344 Fricke, 422 B Fried, 1 Baumslag, 393, 406, 426 Friedl, 425 Benois, 451 Berstel, 361, 451 G Bianchi, 426 Gersten, 324 Binz, 322 Gildenhuys, 216Ð218, 322, 323 Bogopolski, 426, 427 Gilman, 451 Bourbaki, 72, 193, 264 Gitik, 423, 424 Brunner, 424 Goryaga, 427 Burillo, 426 Gregorac, 424 Burns, 323, 423 Gruenberg, 251, 383 Burnside, 105 Grunewald, 344 Bux, 426, 427 Guralnick, 217, 227

C H Chagas, 424, 426 Hall, M., 105, 121, 323, 359, 423 Cohen, 426 Haran, 217, 218, 227, 323, 324 Héam, 425 Coulbois, 423 Henckell, 363, 425 Herfort, 217, 218, 227, 322Ð324 D Herwig, 423 De Oliveira, 424 Hopﬁan, 6 Delgado, 425 Howson, 445 Dicks, 207, 216, 445 Huppert, 116, 264 Doniz, 424 Dunwoody, 207, 216, 445 J Dyer, 323, 426 Jarden, 1, 218

© Springer International Publishing AG 2017 463 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0 464 Index of Authors

K Ribes, 1, 206, 216Ð218, 227, 322Ð324, 381, Kaloujnine, 58 423, 425Ð427 Kargapolov, 58 Rips, 425 Karrass, 371, 383, 395, 422 Romanovskii, 423 Kharlampovich, 426 Rosenberger, 422 Kim, 425 Kochloukova, 324, 426 S Krassner, 58 Sapir, 358, 425 Scheiderer, 221, 323, 324 L Schupp, 248, 426 Lascar, 423 Scott, 323, 424, 426 Lennox, 425 Segal, 343, 344, 379, 381, 395, 425 Lim, 218 Seifert, 361 Lioutikova, 426 Serre, 1, 77, 78, 88, 179, 195, 207, 208, 211, Lubotzky, 218, 323 213, 214, 216, 217, 221, 234, 322, 323, Lyndon, 248, 419, 426 373, 385, 387, 396Ð399, 406, 407, 445 Shusterman, 323 M Solitar, 371, 383, 395, 422, 424 Magnus, 371, 383, 395, 422 Stallings, 423, 441, 445 Mal’cev, 423, 425 Stebe, 422, 425, 426 Margolis, 358, 363, 425 Steinberg, 322, 423, 425, 451 Martino, 426, 427 Swan, 347 McKinsey, 423 T Mel’nikov, 217, 218, 322, 323 Talelli, 425, 426 Merzljakov, 58 Tang, 424Ð426 Metaftsis, 425 Taylor, 426 Minasyan, 427 Tits, 216, 322 Mostowski, 423 Toinet, 426 Myasnikov, 419, 420, 426 V N Varsos, 425, 426 Neukirch, 218, 322 Ventura, 324 Niblo, 423 W O Weil, 217, 358, 425 Oltikar, 322 Wenzel, 322 Wilson, 1, 425, 426 P Wilton, 425, 426 Pin, 363, 423, 425, 451 Wise, 424 Pop, 218, 322 Y R You, 424 Raptis, 425, 426 Remeslennikov, 344, 419, 420, 426 Z Reutenauer, 423, 425 Zalesskii, 1, 217, 218, 322, 324, 381, 423Ð427 Rhodes, 363, 423, 425 Zhou, 425 Index of Terms

A B Action Bianchi group free action on a space, 7 conjugacy separability, 426 of a group on a space, 6 Bockstein homomorphism, 291 Action of a profinite group, 41 Boolean graph, 216 faithful, 119 Bouquet of loops, 63 free, 313 free action, 42 C G-map, 42 C-covering, 74 homology sequence, 265 universal, 74 irreducible, 119 C-standard graph kernel, 41, 119 uniqueness, 196 minimal G-invariant subtrees, 57 C-tree, 49 quotient graph, 42 C-universal covering of a graph of pro-C stabilizer, 42 groups, 195 Additive functor, 17 Cayley graph Admissible pair, 350 of a free group, 61 Algorithms for monoids, 359 of a profinite group, 39 Algorithms in free groups of an abstract group, 435 closure of subgroup, 357 Centralizer, 1 subgroups of finite index, 349 Chain, 56, 433 Alphabet, 359 that equals the full π-tree, 57 Amalgamated product Chain complex of a graph, 45 cyclic amalgamation, 392 Circuit, 41, 431 MayerÐVietoris sequence, 269 Class X , 400 proper, 11, 206, 392 amalgamated products, 418 Augmentation Clopen, 4 ideal, 22 Closed path, 431 map, 22 Closed subgroups Augmentation ideal infinitely generated, 336 and free groups, 59 Coboundary, 22 Automaton, 447 Cocycle, 22 Automorphism, 31 Cofinal, 3 Automorphism of a free pro-p group subsystem, 3 fixed subgroup: of finite rank, 318, 324 Cohomological p-dimension, 26 fixed subgroup: of infinite rank, 318 Cohomology group Axis of a hyperbolic element, 240 of a group, 20

© Springer International Publishing AG 2017 465 L. Ribes, Proﬁnite Graphs and Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 66, DOI 10.1007/978-3-319-61199-0 466 Index of Terms

Coinduced module, 25 Distance, 239 Coinflation, 24 Collapsing of a subgraph, 31 E Compatible pair of maps, 64 Edge Completion origin of an, 29 C pro- , 8 terminus of an, 29 pro-p, 8 Edge group, 178 profinite, 8 Epimorphism Complex of modules, 279 of Galois coverings, 64 splitting, 279 of profinite graphs, 31 Computation of pro-C closure, 355 Equivalence relation Computation of pro-p closure, 357 G-invariant, 43 Cone of X, 79 open, 33 Conjugacy C-separability, 332 Equivariant mapping, 64 Conjugacy distinguished, 333 and amalgamated products, 415 Étale topology and direct products, 424 on Subgp(G), 145 in free-by-C groups, 389 in polycyclic-by-finite groups, 345 F subgroup, 389, 390 Faithful action, 119 Conjugacy separable, 332 Fiber of a sheaf, 137 amalgamated products of groups in X , 418 Filtered from below (collection of subsets), 42 and amalgamated products, 402, 405 Finite state automaton, 447 and direct products, 424 Fixed subgroup of an automorphism, 318 and polycyclic-by-finite groups, 344 Fixed submodule, 20 Bianchi groups, 426 Folding, 441 equivalent conditions, 332 Frattini subgroup, 25 Fuchsian group, 422 Frattini subgroup of a free product, 231 hereditarily, 427 Free iterated amalgamated products, 418 pro-C group, 9 Lyndon group, 420 pro-C product, 10 surface group, 418, 419 product with amalgamation, 11 vs subgroup separable, 424 profinite module, 12 Connected profinite component Free action on graph, 99 of a profinite graph, 38 Free pro-C group Connected profinite graph as a free pro-C product, 168 not abstractly connected, 37 Free pro-C product Connecting homomorphism, 18, 19 counterexample to Kurosh theorem, 313 Constant sheaf over a space, 168 Kurosh subgroup theorem, 232, 275 Continuously indexed subgroups, 145 Converge to 1, 144 of a finite number of groups, 141 Core, 235 of a set of subgroups converging to 1, 144 Core of a subgroup, 281 of a sheaf, 139 Corestriction induced by a family of open subgroup, 233, 235 subgroups, 262 prosolvable subgroup, 322 Covering, 430 restricted, 144 Cycle, 431 Free pro-p product Cyclic subgroup separable, 405 counterexample to Kurosh theorem, 316 Free product D change of pseudovariety, 164 Dihedral pro-σ group, 121 free proabelian product, 167 Direct sum indexed by a topological space minimal trees, 250 of proabelian groups, 167 nonabelianess, 164 of profinite modules, 257 of a constant sheaf, 167 Index of Terms 467

Free-by-C group, 242, 333 connected profinite graph, 36 product of subgroups, 339 C-tree, 49 tree associated, 242 group acting on a profinite graph, 41 Free-by-finite group, 333 incidence maps, 29, 429 product of subgroups, 342 map of graphs, 429 subgroup separability, 333 maximal tree, 433 Frobenius profinite group, 121 morphism, 429 Fuchsian group, 422 path, 431 conjugacy separability, 422 profinite graph, 29 Functor, 13 star of a vertex, 430 additive, 17 the Cayley graph as a G-graph, 44 exact, 13 valency of a vertex, 430 right exact, 13 Graph of groups Fundamental group abstract case, 207 of a graph of groups, 180 C-standard graph of a graph of groups, 195 over a simply connected graph, 182 injective, 206 of a graph of groups, abstract case, 207 MayerÐVietoris sequence, 268 of a graph of groups (construction and reduced, 227 existence), 180 Graph of pro-C groups of a profinite graph, 75 reduced, 227 of an abstract graph, 432 special type, 230 uniqueness, 185, 189 Graph of pro-C groups, 177 C Fundamental pro- group, 75 Graph of profinite groups, 177 is projective, 96 Group conjugacy C-separable, 332 G conjugacy separable, 332 G-graph, 41 free-by-C, 333 connectedness in terms of generators free-by-finite, 333 of G, 44 Frobenius profinite, 121 decomposition as inverse limit, 43 infinite dihedral pro-σ , 121 G-map, 42 nilpotent-by-finite, 343 G-module, 15 G-section, 79 polycyclic, 343 G-space, 6 polycyclic-by-finite, 343 G-transversal, 79 profinite, 4 C 0-transversal, 79 residually , 8 C Galois C-covering, 74 subgroup conjugacy -separable, 333 universal, 74 subgroup conjugacy separable, 333 Galois covering, 63 subgroup separable, 331 associated group, 63 virtually free, 333 compatible pair of maps, 64 Group algebra, 15 connected, 63 complete, 15 finite, 63 Group ring, 15 morphism, 64, 69 Generators H converging to 1, 6 Hall subgroup, 121 Graph Hirsch number, 343 abstract, 429 HNN extension abstract connected, 432 generalized, 184 abstract tree, 432 proper, 207 abstract universal covering, 436 Homology group action of a group, 429 of a group, 24 boolean, 216 Homology groups Cayley graph, 39 of a profinite graph, 46 468 Index of Terms

Homology sequence of action on tree, 265 M Homotopy, 432 M. Hall group, 323 Hyperbolic element, 239 free products, 323 Mal’cev product, 366 I Map Idempotent element, 364 middle linear, 16 Immersion, 430 Map of profinite graphs, 31 failure to be an immersion, 430 Maschke’s Lemma for free pro-p groups, 319 Incidence maps, 29 Maximal tree, 433 for e−1, 29 MayerÐVietoris sequence, 267 Minimal G-invariant subtrees, 57 Induced module, 25 Minimal invariant subtrees, 57 Induced profinite topology on a subgroup, 331 Minimal subtrees, 57, 237 Inflation Module in cohomology, 23 coinduced, 25 Injective free profinite Λ-module, 12 module, 14 G-module, 15 Injective graph of groups, 206 induced, 25 Invariant under an action injective, 14 G-invariant, 57 projective, 14 Inverse Modules limit, 2 direct sum, 257 projection map of inverse limit, 2 Monoid, 359 system, 2 algorithms, 359 Inverse limit of graphs, 33 free, 359 Inverse system of graphs, 33 morphism, 359 Irreducible action on a π-tree, 119 Monomorphism Isomorphism of profinite graphs, 31 of profinite graphs, 31 Morphism, 429 of Galois coverings, 64 K surjective, 64 Kernel Morphism of graphs, 31 of finite monoid, 365 qmorphism, monomorphism, epimorphism, of the action of a profinite group, 41 isomorphism, automorphism, 31 Kleene’s Theorem, 360 Morphism of monoids, 359 Künneth formula, 311 Kurosh subgroup theorem, 232, 273, 275 N n counterexample, 313, 316 -product subgroup separable, 338 Nilpotent-by-finite, 343 Normalizer, 1 L in amalgamated products, 396 Language on an alphabet, 359 in fundamental groups, 397 LERF, 331 Lifting O of a path, 435 Open question, 219, 322, 324, 325, 358, 359, of C-simply connected profinite 424Ð426 subgraphs, 92 Orbit, 42 quotient graphs with no lifting of trees or Order simply connected subgraphs, 78 of a profinite group, 8 Lifting graph, 78 Lifting morphism, 74 P Loop, 29, 429 p-kernel Lyndon group, 419 of finite monoid, 365 conjugacy separability of, 420 p-primary abelian group, 25 Index of Terms 469 p-tree, 51 vertex of a, 29 π-group, 111 vertex set of a, 29 π-tree, 51 Profinite group, 4 and inverse limits of finite trees, 52 acting on a profinite graph, 41 which equals its infinite chains, 57 freely, 42 Path, 29 associated with Galois covering, 63 cycle, 431 Profinite ring, 9, 45 length of a, 29 p-adic integers, 9 on an abstract graph, 431 Profinite space, 3 reduced, 431 G-space, 6 underlying graph, 29 Profinite tree, 51 Polycyclic, 343 Projection Polycyclic-by-finite, 343 of an inverse limit, 2 conjugacy separability, 344 Projective, 99 exactness of completion, 344 group, 10 product of subgroups, 346 module, 14 subgroup conjugacy distinguished, 345 Projective limit of graphs, 33 subgroup conjugacy separability, 344 Projective profinite group, 87 subgroup separability, 343 C-projective, 87 Pontryagin dual, 24 fundamental pro-C group, 96 Poset, 2 Pronilpotent closure, 357 cofinal subset, 3 Pro-p closure, 357 Potent, 394, 426 Pro-p tree, 217 H -potent, 394 Proper quasi-potent, 394 amalgamated profinite product, 11 weakly-potent, 426 Pro-π group, 111 Pro-C closure, 358 Pro-π tree, 217 Pro-C topology Pseudoconstant sheaf over a pointed space, 167 on a free monoid, 363 Pseudovariety Pro-C topology of a group closed under extensions with abelian full, 8 kernel, 57 Pro-C tree, 217 of finite monoids, 366 Product of subgroups in an abstract group, 338 decidable, 366 Product subgroup separable, 338 Pseudovariety of finite groups, 5 Profinite G-space, 45, 102 C, 5 Profinite graph, 29 closed under extensions, 5 collapsing of a subgraph, 31 closed under extensions with abelian connected, 36 kernel, 5 infinite with only finite proper not extension-closed, 58 subgraphs, 40 Pseudovariety of finite monoids connected profinite component, 38 Mal’cev product, 366 C-simply connected, 75 C-tree, 49 Q edge of a, 29 Qmorphism fundamental pro-C group, 75 of profinite graphs, 31 group acting on a profinite graph, 41 Quasi-morphism incidence maps, 29 of profinite graphs, 31 map, 31 Quasi-potent, 394 morphism, 31 Quotient graph, 32 π-tree, 51 under the action of a group, 42 p-tree, 51 qmorphism, 31 R quasi-morphism, 31 Rank of a free pro-C group, 9 subgraph, 31 Rational expression, 360, 361 470 Index of Terms

Rational subset Spanning subgraph, 77 characterization, 360 graph with no C-simply connected Rational subsets of free groups, 449 spanning subgraph, 78 Reduced graph of groups, 227 Specialization, 178 Residually C group, 8 universal, 180 Residually p group, 8 Split complex, 279 Residually solvable group, 8 Stabilizer of an element, 42 Resolution Standard graph, 193 injective, 18 of a graph of pro-C groups, 195 projective, 17 uniqueness, 196 Restricted free pro-C product, 144 Star of a vertex, 430 Restriction Subgraph, 429 in cohomology, 23 collapsing, 31 Ring of a profinite graph, 31 pro-C, 45 spanning, 77 profinite, 11 Subgroup RZ, 1 conjugacy C-distinguished, 333 subnormal, 120 S Subgroup conjugacy C-separable, 333 Second axiom of countability, 4 Subgroup conjugacy distinguished Section, 79 in a free-by-C group, 389 0-section, 79 in a free-by-finite group, 390 existence for C-coverings, 79 in a polycyclic-by-finite groups, 345 fundamental, 79 Subgroup conjugacy separable, 333 Sequence and polycyclic-by-finite groups, 344 exact, 13 Subgroup separable, 331 short exact sequence, 13 and direct products, 424 Shapiro’s Lemma, 25 and free products, 423 Sheaf and nilpotent-by-finite groups, 344 constant, 138, 167 and polycyclic-by-finite groups, 343 constant sheaf, 167 cyclic, 405 epimorphism of sheaves, 138 cyclic subgroup separable, 405 fiber, 137 free product, 423 monomorphism of sheaves, 138 not conjugacy separable, 424 morphism of sheaves, 138 Subgroups morphism to a group, 139 continuously indexed, 145 of groups converging to 1, 138 Submodule of pro-C groups, 137 of fixed points, 20 of profinite modules, 257 Submonoid, 359 pseudoconstant sheaf, 167 Supernatural number, 8 quotient sheaf, 138 Surface group, 418 subsheaf, 138 conjugacy separability of, 418, 419 vertex subsheaf, 177 Simply connected profinite graph, 75 T and profinite trees, 100 Tensor A-completion of a group, 419 finite tree, 76, 87 Tensor induced G-action, 284 vs profinite tree, 76 Tensor product Space complete, 16 profinite, 3 Tensor product induction, 281 second countable, 4 Tensor product of complexes, 279, 280 totally disconnected, 3 Tits line, 239 weight of a, 4 and free products, 240, 250 Spanning C-simply connected subgraph, 77 graphs of residually finite groups, 245 Spanning C-subtree, 77 Trace map, 301 Index of Terms 471

Transversal, 79 V 0-transversal, 79 Valency of a vertex, 430 Tree, 432 Variety, 5 C-tree, 48, 49 Verbal subgroup, 336 minimal G-invariant π-subtree, 57 Vertex π-tree, 48 initial, 29 π-tree depends on π, 105 terminal, 29 π-tree vs simple connectivity, 104 Vertex group, 178 C pro- tree, 217 Virtual p-cohomological dimension, 290 proﬁnite, 51 Virtually free group, 242, 333 tree associated, 242 U Virtually nilpotent group, 343 Underlying graph P of a path, 431 Virtually , 242 Universal covering Virtually polycyclic group, 343 commutes with lim, 75 ←− W construction of a, 82 of a graph of groups, 195 Weakly-potent, 426 Universal covering graph Weight, 153 of an abstract graph, 436 Weight of a topological space, 4, 154 Universal covering of the graph of pro-C Word on an alphabet, 359 groups, 195 Universal coverings Z existence of, 87 Zassenhaus group, 121