The Kurosh Subgroup Theorem for Profinite Groups

UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIÊNCIAS EXATAS DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA

The Kurosh Subgroup Theorem for proﬁnite groups

Mattheus Pereira da Silva Aguiar

Belo Horizonte - MG July 2019 UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIÊNCIAS EXATAS DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA

Mattheus Pereira da Silva Aguiar Supervisor: PhD. John William MacQuarrie

The Kurosh Subgroup Theorem for proﬁnite groups

Thesis submitted to the examination board assigned by the Graduate Program in Mathematics of the Institute of Exact Sciences - ICEX of the Federal University of Minas Gerais, as a partial requirement to obtain a Master Degree in Mathematics.

Belo Horizonte - MG July 2019 “First, have a deﬁnite, clear practical ideal; a goal, an objective. Second, have the necessary means to achieve your ends; wisdom, money, materials, and methods. Third, adjust all your means to that end.”. Aristotle ABSTRACT

A profinite graph Γ is a profinite space with a graph structure, i.e., a closed set named the vertex set and two continuous incidence maps d0, d1 ∶ Γ → V (Γ). A profinite group G can act on a profinite graph and GΓ is called the quotient graph by the action of G. If G acts freely, the map ζ ∶ Γ → GΓ is called a Galois covering of the profinite graph Γ and the group G the associated group of this Galois covering. We can also define a universal Galois covering, where the associated group with this covering is the fundamental group of the profinite graph Γ, C denoted by π1 (Γ). We give an original proof of the Nielsen-Schreier Theorem for free profinite groups over a finite space, which states that every open profinite subgroup of a free profinite group on a finite space is a free profinite subgroup on a finite space using these aforementioned structures. We also define a sheaf of pro-C groups, the free pro-C product of a sheaf and use it to define a graph of pro-C groups and the fundamental group of a graph of pro-C groups in a similar manner we did previously. The analogue of the universal Galois covering is named the C standard graph, denoted by S (G, Γ). Finally we show that the free product of pro-C groups can be seen as the fundamental group of a graph of pro-C groups and use this fact to prove the Kurosh Subgroup Theorem for profinite groups, the main result of this thesis.

Keywords: Proﬁnite graphs, Proﬁnite groups, Galois coverings, Fundamental group, Nielsen- Schreier, Graph of pro-C groups, Kurosh. RESUMO EXPANDIDO

A teoria de Bass-Serre para grafos abstratos foi inaugurada no livro ’Arbres, amalgames,

SL2’ (’Trees’ na versão em inglês), escrito por Jean-Pierre Serre em colaboração com Hyman Bass (1977). A motivação original do Serre era entender a estrutura de certos grupos algébricos cujas construções Bruhat-Tits são árvores. No entanto, a teoria rapidamente se tornou uma ferramenta padrão para a teoria geométrica de grupos e a topologia geométrica, particularmente no estudo de 3-variedades. Um trabalho posterior de Bass contribuiu substancialmente para a formalização e o desenvolvimento das ferramentas básicas e atualmente o termo teoria de Bass-Serre é amplamente empregado para descrever a disciplina. Essa teoria busca explorar e generalizar as propriedades de duas construções já bem conhecidas da teoria grupos: produto livre com amalgamação e extensões HNN. No entanto, ao contrário dos estudos algébricos tradicionais dessas duas estruturas, a teoria de Bass-Serre utiliza a linguagem geométrica de espaços de recobrimento e grupos fundamentais. O análogo profinito iniciou em Gildenhuys e Ribes (1978) onde o nome booleano foi utilizado. O objetivo era construir um paralelo entre a teoria de Bass-Serre de grupos abstratos agindo sobre árvores abstratas para grupos profinitos e aplicações em grupos abstratos. O C- recobrimento galoisiano universal de um grafo conexo também foi definido nesse artigo de Gildenhuys e Ribes (1978). Haran (1987) e Mel’nikov (1989) expandiram essas ideias de maneira independente e desenvolveram abordagens gerais para produtos livres de grupos profinitos indexados por um espaço profinito. O objetivo deles era ser capaz de descrever a estrutura de pelo menos certos subgrupos fechados de produtos pro-p livres de grupos pro-p, demonstrando uma versão profinita do Teorema do Subgrupo de Kurosh, que é o teorema principal dessa dissertação. Os artigos de Haran e Mel’nikov obtém resultados similares. Nesse texto, adotamos a elegante versão de Mel’nikov. As primeiras seções do capítulo 4 são baseadas em Zalesskii e Mel’nikov (1989). O primeiro capítulo dessa dissertação estabelece esse contexto histórico do desenvolvimento da teoria de grafos profinitos. O segundo inicia com a definição de limite inverso com algumas de suas propriedades, o que nos possibilita definir um espaço profinito, que surge como o limite inverso de espaços topológicos finitos, cada um com a topologia discreta. Fornecidas tais definições básicas, definimos um grafo profinito, que é um espaço profinito com estrutura de grafo. O espaço de vértices é um subspaço fechado (logo também profinito) e as aplicações de incidência são aplicações contínuas (no ponto de vista topológico), do espaço todo para o espaço dos vértices. Então definimos um q-morfismo de grafos profinitos, que é uma generalização do morfismo usual de grafos abstratos, pois permite que arestas sejam mapeadas em vértices. Em seguida definimos ação de grupos profinitos em garfos profinitos e o grafo quociente pela ação de um grupo profinito G, que é o espaço das órbitas pela ação de G e como um exemplo construímos o grafo de Cayley de um grupo profinito. No terceiro capítulo introduzimos o conceito de recobrimento galoisiano e do C- recobrimento galoisiano, que é uma aplicação ζ ∶ Γ → ∆ = GΓ de um grafo profinito Γ no grafo quociente pela ação do grupo profinito G, ∆. O grupo profinito G é dito o grupo associado ao recobrimento galoisiano. Definimos então morfismos de recobrimentos galoisianos e o conceito C de C-recobrimento galoisiano universal, cujo grupo associado é o grupo fundamental π1 (Γ) do grafo conexo Γ. Em seguida estabelecemos condições para que 0-seções e 0-transversais existam e a construção dos recobrimentos galoisianos universais. Terminamos o capítulo apresentando dois exemplos importantes: 3.4.5, 3.4.6 e uma demonstração original da versão profinita do Teorema de Nielsen-Schreier (quando o espaço profinito é finito), cujo enunciado estabelece que todo subgrupo aberto de um grupo profinito livre é profinito livre em um espaço finito. Esse teorema é demonstrado em [2], do artigo de Binz, Neukirch e Wenzel [1971] e utiliza a versão abstrata. Ribes e Steinberg (2010) deram uma nova demonstração sem utilizar a versão abstrata, através de produtos entrtelaçados, mas não é tão simples quanto a apresentada nessa dissertação. A abordagem também é completamente diferente. O último capítulo contém os principais tópicos dessa dissertação: a definição de um feixe de pro-C grupos, o produto livre de um feixe de pro-C grupos, grafos de pro-C grupos e especializações, o grupo fundamental de um grafo de pro-C grupos, o grafo padrão de um grafo de pro-C grupos e o Teorema de Kurosh, que estabelece:

Theorem 1 (Kurosh). Seja C uma pseudo-variedade de grupos ﬁnitos fechada para extensão. n Seja G = ∐i=1 Gi o produto pro-C livre de um número ﬁnito de grupos pro-C Gi. Se H é um subgrupo aberto de G, então

n −1 H = O O (H ∩ gi,τ Gigi,τ ) F i=1 τ∈HG~Gi

−1 é um produto pro-C livre de grupos H ∩ gi,τ Gigi,τ , onde, cada i = 1, ⋯, n, gi,τ varia sobre um sistema de representantes das classes laterais duplas HG~Gi, e F é um grupo pro-C livre de posto ﬁnito rF , n rF = 1 − t + Q(t − ti), i=1 onde t = [G ∶ H] e ti = SHG~GiS.

A ideia da demonstração é baseada em enxergar o produto livre de um número ﬁnito de grupos pro-C como o grupo fundamental de um grafo de pro-C grupos em que todos os grupos de aresta são triviais e H age sobre o grafo padrão de pro-C grupos. ACKNOWLEDGEMENT

I would first like to thank God for the life, health, faith, determination and wisdom. I would also like to thank my mother and my sister that always had been on my side encouraging, supporting and giving me strength to surpass all the challenges I had to face in this journey. I love you both. To Professor Anderson Porto that was and is a friendly encouraging, since the mathematical Olympiads, giving me the hand and the road map to achieve uncountable many realisations. Also to Professors Leonardo Gomes and Ana Cristina Vieira for the guidance and to the master and undergraduate colleagues, in special to Maria Cecíllia Alecrim, Moacir Aloísio, José Alves and Mateus Figueira for the companionship and friendship. Finally I would like to thank my thesis advisor Professor John MacQuarrie. The door to Prof. MacQuarrie’s office was always open whenever I had a question or a new idea about my research or writing. The present work was realised with the support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Code of financing 001.

Contents

1 Introduction1

2 Profinite graphs3 2.1 Inverse limits...... 3 2.2 Profinite graphs...... 7 2.3 Groups acting on profinite graphs...... 16

3 The fundamental group of a proﬁnite graph 21 3.1 Galois coverings...... 21 3.2 Universal Galois coverings and fundamental groups...... 26 3.3 0-transversals and 0-sections...... 28 3.4 Existence of universal Galois coverings and the Nielsen-Schreier Theorem for pro-C groups...... 34

4 Graphs of pro-C groups 39 4.1 Free pro-C product of a sheaf of pro-C groups...... 39 4.2 Graphs of pro-C groups and specialisations...... 45 4.3 The fundamental group of a graph of pro-C groups...... 47 4.4 The standard graph of a graph of pro-C groups...... 51 4.5 The Kurosh Theorem for free pro-C products...... 57

Bibliography 61

Chapter 1

Introduction

Bass–Serre theory for abstract graphs was presented in the book ’Arbres, amalgames, SL2’ (Trees in the english version), written by Jean-Pierre Serre in collaboration with Hyman Bass (1977). Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. A subsequent work of Bass contributed substantially to the formalization and development of the basic tools of the theory and currently the term ’Bass–Serre theory’ is widely used to describe the subject. This theory builds on exploiting and generalizing the properties of two older group-theoretic constructions: free products with amalgamation and HNN-extensions. However, unlike the traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language of covering theory and fundamental groups. The notion of a profinite graph first appeared in Gildenhuys and Ribes (1978) where the name boolean was used. The aim was to develop a parallel to the Bass-Serre theory of abstract groups acting on abstract trees for profinite groups and applications to abstract groups. Their work also introduces the universal Galois C-covering of a connected profinite graph, the concept of a ’sheaf of pro-C groups’ and the corresponding free pro-C product. Later, Zalesskii and Mel’nikov (1989) developed the concept of a graph of pro-C groups, the fundamental group of a graph of pro-C groups and the standard graph. These resources will be fundamental to prove the main theorem of this thesis. Chapter 2 provides a review of some important results of inverse limits, profinite spaces and pro-C groups. There is also an introduction to the concept of a profinite graph, that arises as an inverse limit of finite graphs, the action of a profinite group on a profinite graph and the Cayley graph of a profinite group. Chapter 3 starts with the definition of a Galois covering of a profinite graph Γ and the profinite group associated with this Galois covering. Later this group is seen as a subgroup of Aut(Γ). A profinite graph Γ also has a universal Galois covering, constructed by a universal property. As the profinite group G has to act freely on Γ˜ to be a Galois covering of a graph Γ,

1 it admits a section j and a G-transversal J, by Lemma 3.3.9. We finish the chapter giving an idea of the construction of universal Galois coverings, summarized in Construction 3.4.1 and Theorem 3.4.3. We also give two important examples: 3.4.5, 3.4.6 and an original proof of the Nielsen-Schreier Theorem for free profinite groups on a finite space, which states that every open profinite subgroup of a free profinite group on a finite space is a free profinite subgroup on a finite space. This Theorem is proved in [2], from the article of Binz, Neukirch and Wenzel [1971] and uses the abstract version. Ribes and Steinberg (2010) gave a new proof without using the abstract version, through wreath products, but it is not so simple as the one presented in this thesis. The approach is entirely different as well. The last chapter contains the main topics of this thesis: the definition of a sheaf of pro-C groups, the free product of a sheaf of pro-C groups, graphs of pro-C groups and specialisations, the fundamental group of a graph of pro-C groups, the standard graph of a graph of pro-C groups and the Kurosh theorem for profinite groups that states:

Theorem 2 (Kurosh). Let C be an extension-closed pseudovariety of ﬁnite groups. Let G = n ∐i=1 Gi be a free pro-C product of a ﬁnite number of pro-C groups Gi. If H is an open subgroup of G, then n −1 H = O O (H ∩ gi,τ Gigi,τ ) F i=1 τ∈HG~Gi

−1 is a free pro-C 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 HG~Gi, and F is a free pro-C group of ﬁnite rank rF , n rF = 1 − t + Q(t − ti), i=1 where t = [G ∶ H] and ti = SHG~GiS.

The idea of the proof is based on seen the free product of finitely many pro-C groups as the fundamental group of a graph of pro-C groups where all the edge groups are trivial and H acts on the standard graph of pro-C groups. The Theorem was originally proved (the abstract case) by A. G. Kurosh in 1934. After that, there were many subsequent proofs, including proofs of Harold W. Kuhn (1952), Saunders Mac Lane (1958) and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions. One of this important generalizations is the profinite case. Haran (1987) and Mel’nikov (1989) independently expanded these ideas and developed 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, proving a profinite version of the Kurosh Subgroup Theorem, also the main theorem of this thesis. The pappers of Haran and Mel’nikov obtain similar group theoretic results. Here we have adopted the rather elegant viewpoint of Mel’nikov. We will omit the proofs of the results that are not used directly in the main result.

2 Chapter 2

Proﬁnite graphs

In this chapter we define the concepts of inverse limit, profinite space and profinite group that are fundamental for the definiton of a profinite graph and the quotient graph by an action of a profinite group. We also present the definitions of a profinite graph, some properties and concepts, the action of a profinite group on a profinite graph and the Cayley graph of a profinite group. Other properties can be found on chapters 1 and 2 of [1] and chapter 1 of [2].

2.1 Inverse limits

In this section we define the concept of inverse limit and establish some of its elementary properties that will be crucial to the definition of a profinite graph and a profinite group. Rather than developing the concept and establishing those properties under the most general conditions, we restrict ourselves to inverse limits of topological spaces or topological groups. The main topics of this section can be found in [1] and [2].

Deﬁnition 2.1.1 (Directed poset). Let I = (I, ⪯) be a set with binary relation ⪯. It is called a directed partially ordered set or a directed poset if the following holds:

(a) i ⪯ i for i ∈ I;

(b) i ⪯ j and j ⪯ i imply i = j for i, j ∈ I;

(d) if i, j ∈ I, there exists some k ∈ I such that i, j ⪯ k. Deﬁnition 2.1.2 (Inverse system). An inverse system of topological spaces over a directed poset

I consists of a collection of topological spaces {Xi} indexed by I and collection of continuous mappings ϕij ∶ Xi → Xj (deﬁned when i ⪰ j) such that the diagrams of the form

ϕik Xi / Xk >

ϕij ϕjk

3 commute whenever they are deﬁned, i.e., whenever i, j, k ∈ I and i ⪰ j ⪰ k. It is denoted by {Xi, ϕij,I}.

Deﬁnition 2.1.3 (Compatible mappings). Let Y be a topological space, {Xi, ϕij,I} an inverse system of topological spaces over a directed poset I and ψi ∶ Y → Xi be a continuous mapping for each i ∈ I. These mappings are said to be compatible if the following diagram commutes:

ψi Y / Xi

ϕij ψj Xj or equivalently, ϕijψi = ψj, whenever i ⪰ j.

Deﬁnition 2.1.4 (Inverse limit). A topological space X together with compatible continuous mappings

ϕi ∶ X → Xi (i ∈ I) 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 and ψi ∶ Y → Xi is a set of compatible continuous mappings, then there is a unique continuous mapping ψ ∶ Y → X such that ϕiψ = ψi for all i ∈ I. We say that ψ is ’induced’ or ’determined’ by the compatible homomorphisms ψi.

The maps ϕi are called projections. The projection maps ϕi are not necessarily surjections. We denote the inverse limit by (X, ϕi), or often simply by X, by abuse of notation.

Proposition 2.1.5 ([2], Proposition 1.1.1). Let {Xi, ϕij,I} be an inverse system of topological spaces over a directed poset I. Then

(a) There exists an inverse limit of the inverse system {Xi, ϕij,I}.

(b) This limit is unique in the following sense: if (X, ϕi) and (Y, ψi) are two inverse limits of the inverse system {Xi, ϕij,I}, then there is a unique homeomorphism ϕ ∶ X → Y such that ψiϕ = ϕi.

4 Proof. See [2], Proposition 1.1.1, page 2.

If {Xi,I} is a collection of topological spaces indexed by a set I, its direct product or cartesian product is the topological space Πi∈I Xi, endowed with the product topology. In the case of topological groups, the group operation is deﬁned coordinatewise. The inverse limit of such a system, denoted by Γ = lim Γi, is the subset of i∈I Γi consisting of those tuples (mi) ←Ði∈I ∏ with ϕij(mi) = mj, whenever i ⪰ j. For the next proposition, we can recall that

Deﬁnition 2.1.6 (Totally disconnected space). A topological space X is said totally disconnected if every point in the space is it own connected component.

So, if we take the inverse system of a special type of topological spaces (Haussdorf, compact and totally disconnected), their inverse limit maintain these properties. It can be seen through the Tychonoff Theorem for the compact part and the fact that a closed subset of a compact space is compact. For the Hausdorff and totally disconnected parts, these are well known results. We state this in a proposition:

Proposition 2.1.7 ([2], Proposition 1.1.3). Let {Xi, ϕij,I} be an inverse system of compact, Hausdorff and totally disconnected topological spaces over the directed set I. Then

lim Xi ←Ði∈I is also a compact, Haussdorf and totally disconnected space.

The next proposition gives to us conditions where this inverse limit is not empty:

Proposition 2.1.8 ([2], Proposition 1.1.4). Let {Xi, ϕij,I} be an inverse system of compact, Haussdorf nonempty topological spaces Xi over the directed set I. Then

lim Xi ←Ði∈I is nonempty. In particular, the inverse limit of an inverse system of nonempty ﬁnite sets with the discrete topology is nonempty.

Proof. See [2], Proposition 1.1.4, page 4.

The next deﬁnition gives a necessary condition to an inverse system on a subset I′ of the directed poset I have the same inverse limit of the equivalent inverse system in I.

′ Definition 2.1.9 (Cofinal). 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 .

5 Proposition 2.1.10 ([2], Proposition 1.1.9). 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 ←Ði∈I ←Ði ∈I ′ . ′ ′

Proof. See [2], Lemma 1.1.9, page 8.

Remark 2.1.11. On all deﬁnitions and propositions above, if X is also a topological group, the continuous mappings ϕij ∶ Xi → Xj are continuous group homomorphisms.

For further properties of inverse limits, see [2], Section 1.1.

Definition 2.1.12 (Profinite space). A profinite space X is a topological space that is the inverse limit lim Xi of finite spaces Xi endowed with the discrete topology. ←Ði

By Proposition 2.1.8, if each finite space Xi is nonempty, then the profinite space X = lim Xi is also nonempty. The next theorem provides a characterization of profinite spaces: ←Ði Theorem 2.1.13 ([2], Theorem 1.1.12). Let X be a topological space. Then the following conditions are equivalent:

(a) X is a proﬁnite space;

(b) X is compact, Hausdorff and totally disconnected;

Proof. See [2], Theorem 1.1.12, page 10.

To deﬁne a pro-C group we need the notion of a pseudovariety. In [2] it is referred as a variety of proﬁnite groups.

Definition 2.1.14 (Pseudovariety of finite groups). A nonempty class of finite groups C is a pseudovariety if it is closed under taking subgroups, quotients and finite direct products. 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.

6 Now we can deﬁne a pro-C group:

Definition 2.1.15 (Pro-C group). Let C be a nonempty class of finite groups. Define a pro-C group G as an inverse limit

G = lim Gi ←Ði∈I of a surjective inverse system {Gi, ϕij,I} of groups Gi in C, where each group Gi is endowed with the discrete topology.

We think of such a pro-C group G as a topological group, whose topology is inherited from the product topology on Πi∈I Gi. If C is the pseudovariety of ﬁnite groups, we call a pro-C group G a proﬁnite group.

2.2 Proﬁnite graphs

In this section we define profinite graphs and the Cayley graph of a profinite group.

Definition 2.2.1 (Profinite graph). A profinite graph is a profinite space Γ with a distinguished closed nonempty subset V (Γ) called the vertex set, E(Γ) = Γ − V (Γ) the edge set and two continuous maps d0, d1 ∶ Γ → V (Γ) whose restrictions to V (Γ) are the identity map idV (Γ). We refer to d0 and d1 as the incidence maps of the graph Γ.

A profinite graph is also an oriented abstract graph without the topology. As the set of vertices is closed, the edges set is open, but it need not to be closed. When this happens, E(Γ) is also compact (and so profinite) and we only need to check the continuity of the incidence maps 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 the symbols e1 = e and e 1. We define the −1 −1 incidence maps for these symbols as follows: d0(e ) = d1(e) and d0(e) = d1(e ). We can define some structures immediately from the abstract Bass-Serre theory that behave exactly in the same way.

Deﬁnition 2.2.2 (Path). Given vertices v and w of Γ, a path pvw from v to w is a ﬁnite sequence ε1 εm ε1 εm e1 , ⋯, em , where m ≥ 0, ei ∈ E(Γ), εi = ±1, (i = 1, ⋯, m) such that d0(e1 ) = v, d1(em ) = w εi εi 1 and d1(ei ) = d0(ei+1 ) for i = 1, ⋯, m − 1. Such a path is said to have length m. +

Note that a path is always meant to be ﬁnite. The underlying graph of the path pvw consists ε1 εm of the edges e1 , ⋯, em and their vertices di(ej)(i = 0, 1; j = 1, ⋯, m).

Deﬁnition 2.2.3. The path pvw is called reduced if whenever ei = ei+1, then εi = εi+1 for all i = 1, ⋯, m − 1. In abstract Bass-Serre theory there is an algorithm to transform any path into a reduced one, called simple reduction of a path. It also holds for this deﬁnition, because it is the immediate analogue, so we can consider all paths as a reduced path, named the reduced form. We can also prove that this reduced form is unique. For more details, see [3].

7 The ﬁrst immediate example of a proﬁnite graph is the following:

Example 2.2.4 ([1], Example 2.1.1(a)). A ﬁnite abstract graph Γ with the discrete topology is a proﬁnite graph.

Next we give a less trivial example and that will be useful to show certain unique properties that arise from the proﬁnite deﬁnition of connectivity:

Example 2.2.5 ([1], Example 2.1.1(b)). Let N = {0, 1, 2, ⋯} and N = {n S n ∈ N} be two copies of the set of the natural numbers (each one with the discrete topology). Define Γ = N ⊍Ñ ⊍{∞} to be the one-point compactification of the space N ⊍ Ñ. Hence Γ is a profinite space because it is compact, Hausdorff and totally disconnected. We can introduce a profinite graph structure into Γ by setting:

• V (Γ) = N ⊍ ∞;

• E(Γ) = Ñ;

• d0(̃n) = n for ̃n ∈ E(Γ) and d0(n) = n for n ∈ V (Γ);

• d1(̃n) = n + 1 for ̃n ∈ E(Γ) and d1(n) = n for n ∈ V (Γ).

0 1 2 3 4 ... ∞ 0 1 2 3

In this case the subset of edges E(Γ) = Ñ = {̃n S n ∈ N} is open, but not closed in Γ. Indeed, we are taking the discrete topology, so Ñ is open, but it is not compact and therefore not closed.

As now we are working with topological spaces, additional conditions on a subset of a profinite graph are needed to constitute a profinite subgraph. It has be a closed subset in order to be profinite. We state this in the following definition:

Definition 2.2.6 (Profinite subgraph). A nonempty closed subset ∆ of a profinite graph Γ is called a profinite subgraph of Γ if whenever m ∈ Γ, then dj(m) ∈ Γ (j = 0, 1).

The major changes from the abstract world start to appear in the next definition, the q- morphism, where we allow edges go to vertices in a map of profinite graphs. This fits very well with inverse limits constructions that are not needed in the abstract case.

Definition 2.2.7 (q-morphism of profinite graphs). Let Γ and ∆ be profinite graphs. A q- morphism or a quasi-morphism of profinite 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.

8 If α is a surjective q-morphism (respectively injective, bijective), 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.

Remark 2.2.8. The relation dj(α(m)) = α(dj(m)), (j = 0, 1; m ∈ Γ) implies that a q- morphism of proﬁnite graphs maps vertices to vertices. Indeed, suppose for contradiction that given m ∈ V (Γ), α(m) ∈ E(∆). Thus, dj(α(m)) ∈ V (∆) because dj ∶ ∆ → V (∆) and α(dj(m)) ∈ E(∆) because dj(m) = m, and djSV (Γ) = idV (Γ). Thus they cannot be equal. Therefore, a q-morphism can send edges to vertices:

Definition 2.2.9 (Profinite quotient graph, [1], Example 2.1.2). Let Γ be a profinite graph and ∆ a profinite subgraph of Γ. We can define a natural continuous map α ∶ Γ → Γ~∆, where Γ~∆ is endowed with the quotient topology, and a profinite graph structure on the space Γ~∆ inherited from Γ as follows:

• V (Γ~∆) = α(V (Γ));

• dj(α(m)) = α(dj(m)) (j = 0, 1); for all m ∈ Γ (note that we only need to deﬁne the vertex set and the incidence maps on the edges, because they are trivial on the vertices). Then α is a q-morphism of proﬁnite graphs and we call Γ~∆ a quotient graph of Γ. We can say that Γ~∆ is obtained from Γ by collapsing ∆ to a point. Observe that α maps any edge of Γ which is in ∆ to a vertex of Γ~∆.

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 q-morphism of graphs. The following construction provides necessary and sufficient conditions for this to happen.

Construction 2.2.10 ([1], 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 q-morphism 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

9 commutes.

(b) If Y admits a proﬁnite graph structure so that ϕ is a q-morphism, then ψϕ is an isomorphism of proﬁnite graphs.

(c) Consequently, there exists a proﬁnite graph structure on Y such that ϕ is a q-morphism 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 di(ϕ(m)) = ϕ(di(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)) .

Let Γϕ = ϕ˜(Γ). Then Γϕ admits a unique graph structure such that ϕ˜ ∶ Γ → Γϕ is a q- ˜ ˜ morphism 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)) and ˜ d1 (ϕ(m), ϕd0(m), ϕd1(m)) = (ϕd1(m), ϕd1(m), ϕd1(m)) ˜ where ϕ˜(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y . We have to check that d0 ∶ Γϕ → V (Γϕ) and ˜ d1 ∶ Γϕ → V (Γϕ) are well deﬁned. Indeed,

(ϕd0(m), ϕd0(m), ϕd0(m)) ∈ ϕ˜(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y and

(ϕd1(m), ϕd1(m), ϕd1(m)) ∈ ϕ˜(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y.

′ ′ If m = m , (m, m ∈ Γ) we have that

ϕ˜(m) = (ϕ(m), ϕd0(m), ϕd1(m)) and ′ ′ ′ ′ ϕ˜(m ) = (ϕ(m ), ϕd0(m ), ϕd1(m )) .

′ ′ ′ But as m = m , d0(m) = d0(m ), d1(m) = d1(m ) and as ϕ is well deﬁned, we ′ ′ ′ have that ϕ(m ) = ϕ(m), ϕd0(m ) = ϕd0(m), ϕd1(m ) = ϕd1(m). Therefore, d0 and d1 are well deﬁned. The map ϕ˜ is also a q-morphism of graphs, because ϕ˜(dj(m)) = (ϕ(dj(m)), ϕd0(dj(m)), ϕd1(dj(m))). As d0 and d1 are the identity on V (Γ),

10 ϕ˜(dj(m)) = (ϕdj(m), ϕdj(m), ϕdj(m)) ˜ = dj (ϕ(m), ϕd0(m), ϕd1(m)) as desired. 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 q-morphism 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 q-morphism of graphs and it is inverse to ψϕ. This proves properties (a) and (b). Property (c) is clear. Property (d) says that E(Γ) is closed (and so compact) and ϕ(E(Γ)) ∩ ϕ(V (Γ)) = ∅. By the continuity of ϕ, ϕ(E(Γ)) is compact in Y and so closed and ϕ(E(Γ)) ∩ ϕ(V (Γ)) = ∅ is the intersection of two clopen sets. Using this information together with the definition of ϕ˜,

ϕ˜(m) = (ϕ(m), ϕd0(m), ϕd1(m)), we conclude that ϕ˜ is a morphism of proﬁnite graphs and as ψϕ is an isomorphism, ψϕ(E(Γϕ)) ∩ ψϕ(V (Γϕ)) = ∅, proving (d).

Definition 2.2.11 (Inverse system of profinite graphs). Let (I, ⪰) be 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 q-morphisms 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 of such a system Γ = lim Γi is the subset of i∈I Γi consisting of those ←Ði∈I ∏ tuples (mi) with ϕij(mi) = mj, whenever i ⪰ j. Therefore, we can define a natural profinite graph structure in the inverse limit by defining V (Γ) = lim V (Γi) and the incidence maps to ←Ði∈I be the composition with the projection maps. 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. We can define 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, given R1,R2 ∈ R there exists R3 ∈ R such that ϕR3 ∶ Γ → Γ~R3 is the quotient map and R3 ⪰ R1,R2. Hence (R, ⪯) is in fact a directed poset and {Γ~R, ϕR1,R2 , R} is an

11 inverse system over R, and, as topological spaces, the collection of quotient maps {ϕR S R ∈ R} induces a homeomorphism from Γ to lim Γ~R; Therefore, we can identify these two spaces ←ÐR∈R by means of this homeomorphism and write

Γ = lim Γ~R. (2.1) R←Ð∈R ′ Consider now the subset R of R consisting of those R ∈ R such that Γ~R admits a unique graph structure by Construction 2.2.10 so that ϕR ∶ Γ → Γ~R is a q-morphism 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. Let ϕR ∶ Γ → Γ~R be the corresponding quotient map. Let Γ and ϕ Γ Γ R be as in Construction 2.2.10. ϕR ÈR ∶ → ~

ϕR Γ / Γ~R (2.2) ;

ϕÈR ψϕ Γ Γ R˜ ϕR = ~

Then Γ Γ R˜, where R˜ is the equivalence relation on Γ whose equivalence classes are ϕR = ~ ϕÈ−1 x x Γ . Clearly R˜ ′ and R˜ R; hence R˜ R ,R , as needed. { R ( )S ∈ ϕR } ∈ R ⪰ ⪰ 1 2

′ Remark 2.2.12. If R1,R2 ∈ R and R1 ⪰ R2, then the map ϕR1,R2 ∶ Γ~R1 → Γ~R2 is in fact a q- morphism 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 q-morphisms over the directed poset R . We summarise this construction in the following proposition:

Proposition 2.2.13 ([1], 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←Ð∈R ′ Consequently V (Γ) = lim V (Γ~R). R←Ð∈R ′ ′ (b) If the subset E(Γ) of edges of Γ is closed, then a directed subposet R” of R can be chosen

so that whenever R1,R2 ∈ R” with R1 ⪰ R2, then ϕR1,R2 ∶ Γ~R1 → Γ~R2 is a morphism of graphs and Γ = lim Γ~R. R←Ð∈R”

12 Consequently,

V (Γ) = lim V (Γ~R) and E(Γ) = lim E(Γ~R). ←ÐR∈R” ←ÐR∈R” Proof. See [1], Proposition 2.1.4, page 34.

The next proposition gives an important tool to work with inverse limits:

Proposition 2.2.14 ([1], Proposition 2.1.5). Let {Γi, ϕij,I} be an inverse system of proﬁnite graphs and q-morphisms over a directed poset I, and set

Γ = lim Γi. (2.3) ←Ði∈I Let ρ ∶ Γ → ∆ be a q-morphism into a ﬁnite graph ∆. Then there exists a k ∈ I such that ′ ′ ρ factors through Γk, i.e., there exists a q-morphism ρ ∶ Γk → ∆ such that ρ = ρ ϕk, where ϕk ∶ Γ → Γk is the projection.

Proof. See [1], Lemma 2.1.5, page 35.

Another very different deﬁnition is the one of connectivity. As we will show, there exists proﬁnite graphs that are connected, but have vertices with no edge beginning or ending at it.

Definition 2.2.15 (Connected profinite graph). A profinite graph Γ is said to be connected if whenever ϕ ∶ Γ → A is a q-morphism of profinite graphs onto a finite graph, then A is connected as an abstract graph.

We can now summarise some of the properties related to connected proﬁnite graphs.

Proposition 2.2.16 ([1], Proposition 2.1.6). (a) Every quotient graph of a connected proﬁnite graph is connected.

(b) If

Γ = lim Γi ←Ði∈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 SΓS > 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 and 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 ˜ −1 projection. Then the inverse image Λ = α (Λ) in Γ of a connected profinite subgraph Λ of Γ~∆ is a connected profinite subgraph.

Proof. See [1], Proposition 2.1.6, page 36.

13 Proposition 2.2.17 ([1], Proposition 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, then D is a connected profinite graph.

(b) Let {∆j S 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. See [1], Proposition 2.1.7, page 37.

The following example shows a connected proﬁnite graph which is not connected as an abstract graph and has a vertex with no edge beginning or ending at it.

Example 2.2.18 ([1], Example 2.1.8). Consider Γ the same graph of Example 2.2.5, Γ = ˜ N ⊍ N ⊍ {∞}, the one point compactiﬁcation of a disjoint union of two copies N and ˜ N = {n˜ S n ∈ N} of the natural numbers;

• V (Γ) = N ⊍ {∞}; ˜ • E(Γ) = N;

• d0(n˜) = n;

• d1(n˜) = n + 1 for n˜ ∈ E(Γ). We can represent Γ as follows:

0 1 2 3 4 ... ∞ 0 1 2 3

Then Γ is a connected proﬁnite graph; to see this consider the connected ﬁnite graphs Gn

0 1 2 3 ... n − 1 n 0˜ 1˜ 2˜ nÊ− 1

˜ ˜ ˜ ˜ with vertices V (Γn) = {0, 1, 2, 3, ⋯, n} and edges E(Γn) = {0, 1, 2, ⋯, nÊ− 1} such that d0(i) = ˜ i, d1(i) = i + 1 (i = 0, 1, ⋯, n − 1) and dj(i) = i for i ∈ V (Γ), (i = 0, 1, ⋯, n; j = 0, 1). If n ≤ m, deﬁne ϕm,n ∶ Γm → Γn 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 {Γn, ϕm,n} is an inverse system of ﬁnite graphs, and

Γ = lim Γn, n←Ð∈N where ∞ = (n)n∈N (note here the importance of the new definition of q-morphism. Here are allowed to send the edge [n, m] to the vertex n, what would not be possible with the abstract definition of a morphism). Hence Γ is a connected profinite graph, because any morphism of G to a finite graph can be factored through a finite connected graph (cf. Proposition 2.2.14), so the finite graph has to be connected. We observe that there is no edge e of Γ which has ∞ as one of its vertices; and so Γ is not connected as an abstract graph.

14 The next proposition is another characterisation of connected profinite graphs and it is easily proved using the quotient profinite graph. We can collapse Γ1 to a point and Γ2 to another point in order to get a disconnected finite quotient graph with two vertices and no edges. Therefore, it cannot be connected.

Proposition 2.2.19 ([1], Proposition 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.

The following is a natural deﬁnition motivated by the topology and the abstract graph theory:

Definition 2.2.20 (Connected profinite component). A maximal connected profinite subgraph of a profinite graph Γ is called a connected profinite component of Γ.

The next proposition characterises these connected proﬁnite components:

Proposition 2.2.21 ([1], 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.

Proof. See [1], Proposition 2.1.10, page 38.

We finish this section with the definition of a Cayley graph of a profinite group, that will be very useful through this thesis:

Definition 2.2.22 (Cayley graph). Let G be a profinite group and X a closed subset of G. Put X̃ = X ∪ {1}. We can define the Cayley graph Γ(G, X) of the profinite group G with respect to the subset X as follows:

• Γ(G, X) = G × X̃

• V (Γ(G, X)) = {(g, 1)S g ∈ G} ˜ • dj ∶ Γ(G, X) = G × X → V (Γ(G, X))

– d0(g, x) = g

– d1(g, x) = gx where G×X̃ has the product topology and j = 0, 1. Note that G×X̃ is a profinite space, because G is a profinite group (and in particular a profinite space), X is a closed subspace of a profinite space, and so it compact and therefore profinite. Hence, G × X̃ with the product topology is also profinite.

15 We can identify the space V (Γ(G, X)) with G through the homeomorphism (g, 1) ↦ g (g ∈ G). The incidence maps d0 and d1 are continuous (one is just a projection and the other is a left multiplication) and they are the identity map when restricted to V (Γ(G, X)) = {(g, 1)S ˜ g ∈ G} = G. Therefore Γ(G, X) = 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 already open, because V (Γ(G, X)) is closed in Γ(G, X). It is also ˜ closed if and only if 1 is an isolated point of X. Indeed, note 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. On the other hand, if 1 ∈ X ˜ we have that X = X. If in addition 1 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. Let ϕ ∶ G → H be a continuous homomorphism of profinite groups and let X be a closed subset of G, and so compact. Put Y = ϕ(X) that is a compact subset of H by the continuity of ϕ and therefore closed. So we can define the Cayley graphs of G with respect to the subset X, denoted by Γ(G, X) and of H with respect to the subset ϕ(X), denoted by Γ(H, ϕ(X)). Then ϕ induces a q-morphism 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 ϕÈU ∶ Γ(G, X) → Γ(G~U, XU ). So we can construct with the morphisms ϕÈU ∶ Γ(G, X) →

Γ(G~U, XU ) and ϕÉUij ∶ Γ(G~U1,XU1 ) → Γ(G~U2,XU2 ) an inverse system such that

Γ(G, X) = lim Γ(G~U, XU ) U←Ð◁oG is a decomposition of Γ(G, X) as an inverse limit of ﬁnite Cayley graphs.

2.3 Groups acting on proﬁnite graphs

In this section we define an action of a profinite group on a profinite graph and the quotient graph by an action of a profinite group. These concepts will be useful to define Galois coverings in the next chapter.

Definition 2.3.1. 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

16 (i) G acts continuously on the topological space Γ on the left, i.e., there is a continuous map G × Γ → Γ, denoted by (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. Next we deﬁne an important topology that will be used on the space of automorphisms of Γ, denoted Aut(Γ), the compact-open topology.

Deﬁnition 2.3.2. The compact-open topology on Aut(Γ) is generated by a sub-base of open sets of the form B(K,U) = {f ∈ Aut(Γ)S f(K) ⊆ U}, where K ranges over the compact subsets of Γ and U ranges over the open subsets of Γ.

So, if a profinite group G acts on a profinite graph Γ, for a fixed g ∈ G, we can define a map ρg ∶ Γ → Γ given by m ↦ gm (m ∈ Γ). This map is an automorphism of the graph Γ and, by (cf. [2], 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 this action is the kernel of ρ, i.e., the closed normal subgroup of G consisting of all elements g ∈ G such that gm = m, for all m ∈ Γ. We can define actions on the right in a similar manner, but we will only consider left actions in this thesis.

Definition 2.3.3 (G-map of graphs). Let G be a profinite group that acts continuously on two ′ ′ profinite graphs Γ and Γ . A q-morphism of graphs ϕ ∶ Γ → Γ is called a G-map of graphs if

ϕ(gm) = gϕ(m) for all m ∈ Γ, g ∈ G. Definition 2.3.4 (Stabiliser). Assume that a profinite group acts on a profinite graph Γ and let m ∈ Γ. Define Gm = {g ∈ G S gm = m} to be the stabiliser, or G-stabiliser of the element m.

It follows from the continuity of the action and the compactness of G that Gm is a closed subgroup of G. We have that

Gm ≤ Gdj (m) for every m ∈ Γ, j = 0, 1, because Gdj (m) = {g ∈ G S gdj(m) = dj(m)} and as gdj(m) = dj(gm),

Gdj (m) = {g ∈ G S dj(gm) = dj(m)}.

17 Deﬁnition 2.3.5 (Free action). If the stabiliser Gm of every element m ∈ Γ is trivial, i.e., Gm = 1, we say that G acts freely on Γ.

Deﬁnition 2.3.6 (G-orbit). If m ∈ Γ, the G-orbit of m is the closed subset Gm = {gm S g ∈ G}. The next deﬁnition is very important for the Galois coverings, that will be presented in the next chapter:

Definition 2.3.7 (Quotient graph under the action of a profinite group). Let G be a profinite group that acts on a profinite graph Γ. In particular, G acts on V (Γ) (also a profinite space, because it is a closed subset of Γ) and E(Γ). The space

GΓ = {Gm S m ∈ Γ} of G-orbits with the quotient topology is a proﬁnite space which admits a natural graph structure as follows:

• V (GΓ) = GV (Γ);

• dj(Gm) = Gdj(m)(j = 0, 1).

The proﬁnite graph GΓ is called the quotient graph of Γ under the action of G. The corresponding quotient map ϕ ∶ Γ → GΓ is an epimorphism of proﬁnite graphs given by m ↦ Gm (m ∈ Γ, g ∈ G).

Remark 2.3.8. The map ϕ sends edges to edges (it is a morphism). Indeed, given e ∈ E(Γ) the element ϕ(e) = Ge belongs to GE(Γ) = G(Γ − V (Γ)) = GΓ − GV (Γ) = GΓ − V (GΓ) = E(GΓ).

Remark 2.3.9. If N ◁c G, there is an induced action of G~N on NΓ deﬁned by

(gN)(Nm) = N(gm) for g ∈ G, m ∈ Γ. Proposition 2.3.10 ([1], Proposition 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←Ð∈N

Then the collection of graphs {NΓ S N ∈ N } is an inverse system in a natural way and

Γ = lim NΓ. N←Ð∈N

18 ′ (b) Let N ◁c G. For m ∈ Γ, denote by m the image of m in NΓ. Consider the natural action of G~N on NΓ 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 NΓ.

Let Γ be a proﬁnite graph. If {Γi, ϕij,I} is an inverse system of proﬁnite G-graphs and G-maps over the directed poset I, then

Γ = lim Γi ←Ði∈I is in a natural way a profinite G-graph, defined by V (Γ) = lim Γi and the incidence maps ←Ði∈I as the compositions. As each Γi is a profinite G-graph, we have that V (Γi) = V (GΓi) = GV (Γi). Therefore, V (Γ) = lim Γi = lim GV (Γi) = G(lim V (Γi)) = GV (Γ), ←Ði∈I ←Ði∈I ←Ði∈I where Γ = lim Γi. ←Ði∈I Proposition 2.3.11 ([1], Proposition 2.2.2). Let a profinite group G act on a profinite graph Γ.

(a) Then there exists a decomposition

Γ = lim Γi ←Ði∈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. See [1], Proposition 2.2.2, page 43.

Example 2.3.12 (The Cayley graph as a G-graph, [1], Example 2.2.3). Let G be a profinite group and X be a closed subset of G and Γ(G, X) the Cayley graph of G with respect to X as defined in Example 2.2.22. Define a left action of G on Γ(G, X) by setting

′ ′ g ⋅ (g, x) = (g g, x)

˜ ′ ∀x ∈ X = X ∪ {1} and g , g ∈ G. We can see that gdi(m) = di(gm), for all g ∈ G, m ∈ Γ(G, X), (i = 0, 1). Indeed, if m ∈ E(Γ(G, X)), we can write m = (g, x) such that d0(g, x) = g and d1(g, x) = gx, so ′ ′ ′ g d0(g, x) = gg = d0(gg , x) and ′ ′ ′ g d1(g, x) = g gx = d1(g g, x). The incidence maps are trivial on the vertices. Hence, G acts (continuously and freely) on the Cayley graph Γ(G, X).

19 Now if N is the collection of all open normal subgroups of G, we have

Γ(G, X) = lim Γ(G~N,XN ), N←Ð∈N where XN is the image of X in G~N. Note that G~N also acts freely on Γ(G~N,XN ).

We finish this chapter with an necessary and sufficient condition for a Cayley graph of a profinite group to be a connected profinite graph.

Proposition 2.3.13 ([1], Proposition 2.2.4). (a) 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.

(b) 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. See [1], Lemma 2.2.4, page 44.

20 Chapter 3

The fundamental group of a proﬁnite graph

In this chapter we define Galois coverings, universal Galois coverings, the fundamental group of a profinite graph, show cases when a q-morphism ϕ ∶ Γ → GΓ of profinite graphs admits a continuous section (and when it does not) and give an idea of the construction of the universal Galois coverings, with some important examples as Example 3.4.5. We finish this chapter with an original proof of the Nielsen-Schreier Theorem for free profinite groups on a finite space using everything stated so far.

3.1 Galois coverings

In this section we deﬁne a Galois covering of a proﬁnite graph, the associated group with this Galois covering and obtain some properties.

Definition 3.1.1 (Galois covering). Let G be a profinite group that acts freely on a profinite graph Γ. The natural epimorphism of profinite graphs ζ ∶ Γ → ∆ = GΓ of Γ onto the quotient graph by the action of G, ∆ = GΓ is called a Galois covering of the profinite graph ∆. The associated group G is called the group associated with the Galois covering ζ and we denote it by G = G(ΓS∆). If Γ is finite, one says that the Galois covering ζ is finite. The Galois covering is said to be connected if Γ is connected.

Remark 3.1.2. The Galois covering ζ is always a morphism, i.e., always send edges to edges (cf. Remark 2.3.8) and if Γ is ﬁnite then the associated group G(ΓS∆) is also ﬁnite.

Example 3.1.3 ([1], Example 3.1.1(a)). Let ζ ∶ Γ → ∆ be a Galois covering of the proﬁnite graph ∆ with associated group G = G(ΓS∆). Let K ◁c G. Then G~K acts freely on KΓ and KΓ → ∆ is also a Galois covering of ∆, with associated group G~K (cf. Remark 2.3.9).

Example 3.1.4 ([1], Example 3.1.1(b)). Let (X, ∗) be a pointed profinite space (i.e., X is a profinite space with a distinguished point ∗). Define a profinite graph B = B(X, ∗) by B = X,

21 V (B) = {∗} and di(x) = {∗}, (x ∈ X) for i = 0, 1. This graph B(X, ∗) is named the bouquet of loops associated to (X, ∗). For example, if X has 7 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 2.3.12 is free. Therefore the natural epimorphism

ζ ∶ Γ(G, X) → GΓ(G, X) is a Galois covering. We can see that GΓ(G, X) is just the bouquet of loops B(X ∪ {1}, 1) because all the vertices go to the same orbit with the action deﬁned in Example 2.3.12, that is ′ ′ g (g, x) = (g g, x).

Deﬁnition 3.1.5 (Morphism of Galois coverings). Let ζ1 ∶ Γ1 → ∆1, ζ2 ∶ Γ2 → ∆2 be Galois coverings such that G1 = G(Γ1S∆1) and G2 = G(Γ2S∆2). We can deﬁne a morphism of Galois coverings

ν ∶ ζ1 → ζ2 as a pair ν = (γ, f), where γ ∶ Γ1 → Γ2 is a q-morphism of 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. This morphism ν of Galois coverings induces a unique q-morphism of proﬁnite graphs

δ ∶ ∆1 → ∆2 such that the diagram

γ Γ1 / Γ2

ζ1 ζ2

δ ∆1 / ∆2 commutes. Indeed, for all m ∈ Γ1 and g ∈ G1, gm will be mapped to the same element of ∆1, named a = ζ1(m). By deﬁnition of γ, we have that γ(gm) = f(g)γ(m), so as f(g) ∈ G2 and γ(m) ∈ Γ2 it constitutes a unique orbit of the action of G2 on Γ2. Therefore, ζ2(γ(gm)) is a unique element in ∆2, named b = ζ2(γ(gm)). Now we can deﬁne δ ∶ ∆1 → ∆2 to map δ(a) = b making the diagram commute.

Deﬁnition 3.1.6 (Epimorphims). The morphism ν = (γ, f) is called surjective or an epimorphism if γ and f (and hence δ) are epimorphisms.

22 With the morphisms of Galois coverings deﬁned above we can construct an inverse system

{ζi ∶ Γi → ∆i, νij = (γij, fij),I} indexed by a directed poset I. We stablish some notation as follows: the associated groups of the Galois coverings ζi, G(ΓiS∆i) will be denoted by G(ΓiS∆i) = 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 Gi acts continuously and ←Ð freely on the profinite graph Γ = lim Γi; hence, the quotient profinite graph GΓ is isomorphic ←Ð with the profinite graph ∆ = lim ∆i. ←Ð Therefore we have established the following proposition:

Proposition 3.1.7 ([1], Proposition 3.1.2). The inverse limit lim ζi of Galois coverings ←Ð

ζi ∶ Γi → ∆i with associated group Gi = G(ζi) is a Galois covering with associated group lim Gi. ←Ð A useful property for proﬁnite structures is the decomposition as an inverse limit. It does not always hold, but the next proposition shows that it is true for Galois coverings.

Proposition 3.1.8 ([1], Proposition 3.1.3). Any Galois covering ζ ∶ Γ → ∆ of proﬁnite graphs can be decomposed as an inverse limit of ﬁnite Galois coverings, with surjective projections.

Proof. See [1], Proposition 3.1.3, page 65.

The following proposition gives an equivalent way of viewing morphisms of Galois coverings.

Proposition 3.1.9 ([1], Proposition 3.1.4). Let ζ1 ∶ Γ1 → ∆1, ζ2 ∶ Γ2 → ∆2 be connected Galois coverings with associated groups G1 = G(Γ1S∆1) and G2 = G(Γ2S∆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).

23 Proof. See [1], Proposition 3.1.4, page 67.

The next propositions will be useful for further references.

Proposition 3.1.10 ([1], Proposition 3.1.5). Let ζ ∶ Γ → ∆, ξ ∶ Σ → ∆ be connected Galois coverings and let γ ∶ Γ → Σ be a q-morphism such that ζ = ξγ.

γ Γ / Σ

ζ ξ ∆

Let f ∶ G(ΓS∆) → G(ΣS∆) be the homomorphism constructed in Proposition 3.1.9. Then (a) The maps γ and f are surjective;

(b) The map γ is a Galois covering with G(ΓSΣ) = Ker(f), and consequently

G(ΣS∆) ≅ G(ΓS∆)~G(ΓSΣ)

Proof. See [1], Proposition 3.1.5, page 69.

Corollary 3.1.11 ([1], Corollary 3.1.6). Let

γ Γ1 / Γ2

ζ1 ζ2

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

Proof. See [1], Corollary 3.1.6, page 71.

Proposition 3.1.12 ([1], Proposition 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. See [1], Proposition 3.1.7, page 71.

We can show that it is possible to see G(ΓS∆) as a closed subgroup of Aut(Γ), where Aut(Γ) is endowed with the compact-open topology. Let ζ ∶ Γ → ∆ be a Galois covering of a profinite graph ∆, and G = G(ΓS∆) be the associated profinite group. If we fix an element g of G, it determines a continuous automorphism

νg ∶ Γ → Γ

24 given by νg(m) = gm (g ∈ G, m ∈ Γ), as stated before. Hence, the map

ν ∶ G → Aut(G) which sends g to νg is a homomorphism, and it is injective because G acts freely on Γ. Therefore, the space Aut(Γ) can be endowed with the compact-open topology, in such a way that G(ΓS∆) can be seen as a closed subgroup of Aut(Γ), as the next proposition shows:

Proposition 3.1.13 ([1], Proposition 3.2.1). Let ζ ∶ Γ → ∆ be a Galois covering of a profinite graph ∆, and let G = G(ΓS∆) 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 group, i.e., there exists a topological isomorphism of G with a closed subgroup of Aut(Γ).

Proof. See [1], Proposition 3.2.1, page 72.

Let ζ ∶ Γ → ∆ be a connected Galois covering of a proﬁnite graph ∆, and put G = G(ΓS∆). By the previous proposition, we may think of G as a closed subgroup of Aut(Γ). Let

H = NAut(Γ)(G) be the normalizer of G in Aut(Γ). Then H is a closed subgroup of Aut(Γ) because G is already seen as a closed subgroup of Aut(Γ) and if f ∈ H, then f induces a map

Φ(f) ∶ ∆ = GΓ → ∆ = GΓ deﬁned by Φ(f)(Gm) = Gf(m), (m ∈ Γ).

The next proposition summarises some properties of the induced maps Φ(f) and Φ:

Proposition 3.1.14 ([1], 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.

(d) If ∆ is ﬁnite, then the homomorphism Φ ∶ H → Aut(∆) is open and the group H = NAut(Γ)(G) is proﬁnite.

Proof. See [1], Proposition 3.2.2, page 73.

25 3.2 Universal Galois coverings and fundamental groups

In this section we deﬁne a universal Galois covering and its associated group, the fundamental group of a proﬁnite graph.

Deﬁnition 3.2.1 (Galois C-covering). 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(ΓS∆) is a pro-C group.

Remark 3.2.2. Since a Galois C-covering is a Galois covering, all the results of the previous sections are valid for Galois C-coverings.

Definition 3.2.3 (Universal Galois C-covering). A universal Galois C covering is a connected Galois C-covering that respects the following universal property: given any q-morphism β ∶ Γ → ∆ to a connected profinite graph ∆, any connected Galois C-covering ξ ∶ Σ → ∆, and any points ˜ m ∈ Γ̃, s ∈ Σ such that βζ(m) = ξ(s), there exists a q-morphism of profinite graphs α ∶ Γ → Σ, such that βζ = ξα and α(m) = s α Γ̃ / Σ

ζ ξ

β Γ / ∆ We say that α lifts β, or that α is a lifting (q-morphism) of β.

Remark 3.2.4. Once m ∈ Γ̃ and s ∈ Σ with βζ(m) = ξ(s) are given, the lifting q-morphism α is unique, by Proposition 3.1.12. Indeed, suppose the lifting is not unique. We have then α1 and

α2 such that α1 $ Γ̃ Σ : α2 ζ ξ

β Γ / ∆ and βζ = ξα1,βζ = ξα2. Hence ξα1 = ξα2 and we have, as stated in the lemma, a connected Galois covering ξ, a connected profinite graph Γ̃ and α1, α2 ∶ Γ̃ → Σ q-morphisms of profinite graphs with ξα1 = ξα2 and α1(m) = α2(m) (commutativity of the diagram), for some m ∈ Γ̃, hence α1 = α2 as desired. Note also that if the map β is surjective, so is α by Corollary 3.1.11. Furthermore, it follows from Proposition 3.1.8 that it is sufficient to check the universal property above for finite Galois C-coverings ξ ∶ Σ → ∆.

26 Proposition 3.2.5 (Uniqueness of universal Galois C-coverings, [1], 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. See [1], Proposition 3.3.1, page 75.

Definition 3.2.6 (Fundamental pro-C group of a connected profinite graph). Let ζ ∶ Γ̃ → Γ be the C ˜ universal Galois covering of the connected profinite graph Γ. The pro-C group π1 (Γ) = G(ΓSΓ) is called the fundamental pro-C group of Γ.

Remark 3.2.7. The fundamental pro-C group of Γ is well-deﬁned up to isomorphism. Indeed, C suppose for absurd that there exists another fundamental group π1 (Γ) of Γ, which implies

γ Γ̃ / Γ̃′

ζ ξ Γ

C ′ C such that G(G̃SΓ) = π1 (Γ) and G(G̃ SΓ) = π1 (Γ). By Proposition 3.1.10, γ is a Galois covering (isomorphism by Proposition 3.2.5(b)) with G(Γ̃SΓ̃′) = Ker(f) (the map f being C C f ∶ π1 (Γ) → π1 (Γ)) and ′ C C {1} = G(Γ̃SΓ̃ ) ≅ π1 (Γ)~π1 (Γ).

C C Therefore, π1 (Γ) ≅ π1 (Γ).

Definition 3.2.8 (C-simply connected profinite graph). We say that a connected profinite graph C Γ is C-simply connected if π1 (Γ) = 1.

Next we show that universal Galois C-coverings commutes with inverse limits:

Proposition 3.2.9 ([1], 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. ←Ði∈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 ←Ði∈I

27 ˜ 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 = lim π1 (Γi). ←Ði∈I

Proof. See [1], Proposition 3.3.2, page 75.

The last proposition of this section characterises ﬁnite trees and their inverse limits as C- simply connected proﬁnite graphs.

Proposition 3.2.10 ([1], 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 a C-simply connected profinite graph. Proof. See [1], Proposition 3.3.3, page 76.

3.3 0-transversals and 0-sections

In this section we show that a map ϕ ∶ X → GX, where G is a proﬁnite group and X is a proﬁnite G-space does not always have a continuous section. However, if G acts freely on X, e.g., ϕ is a Galois covering, then Lemma 3.3.9 states that this section always exist. This will be vital for the proof of the Kurosh Subgroup Theorem.

Definition 3.3.1 (Spanning profinite subgraph). 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). Another very interesting example is the following: it is well known that every connected abstract graph has a spanning subtree, but it is not always true for profinite graphs. Indeed, the next example shows a connected profinite graph with no spanning C-simply connected profinite subgraph:

Example 3.3.2 ([1], Example 3.4.1). 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. Define a profinite graph Γ = N × {0, 1} with space of vertices and edges

• V (Γ) = {i = (i, 0)S i ∈ N};

• E(Γ) = {i = (i, 1)S i ∈ N};

• d0(i) = i for ̃n ∈ E(Γ) and d0(i) = i for i ∈ V (Γ);

• d1(i) = n + 1 for i ∈ E(Γ) and d1(i) = i for i ∈ V (Γ). where ∞ + 1 = ∞.

28 So we can represent Γ as follows:

0 1 2 3 4 ... ∞ 0 1 2 3 ∞

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

0 1 2 3 ... n − 1 ∞ 0 1 2 n 1 − ∞ where the canonical map Γ(n + 1) → Γ(n) sends i to i identically, if i ≤ n − 1, and it sends n and ∞ to ∞. Hence Γ is a connected profinite graph. We claim that any connected profinite subgraph Γ′ of Γ coincides with Γ; First note that the profinite graph ∆

• V (∆) = {i = (i, 0)S i ∈ N};

• E(∆) = {i = (i, 1)S i ∈ N};

• d0(i) = i for ̃n ∈ E(∆) and d0(i) = i for i ∈ V (∆);

• d1(i) = n + 1 for i ∈ E(∆) and d1(i) = i for i ∈ V (∆).

0 1 2 3 4 ... ∞ 0 1 2 3 is not a spanning profinite subgraph of Γ, because it is not a closed subset of Γ. Indeed, suppose for absurd that ∆ is closed in Γ. Since V (∆) = V (Γ) and it is a closed subset of Γ = V (Γ)⊍E(Γ), we only have to verify that E(∆) is closed in E(Γ). But E(Γ) is closed in Γ (because it is compact), so it is profinite. Thus, E(∆) must be compact, an absurd. Therefore, ∆ is not a profinite subgraph of Γ. To prove the claim, since Γ′ is connected and contains all the vertices of Γ, it must contain ′ all the edges of the form i (i = 0, 1, ⋯); therefore since Γ is compact, it also contains ∞; this proves the claim. On the other hand, if C is a pseudovariety of finite groups, we see that C π1 (Γ) ≅ ZCˆ (cf. Example 3.4.6). Hence Γ does not contain any spanning C-simply connected profinite subgraph.

Definition 3.3.3 (Lifting). Let G be a profinite group that acts on a connected profinite graph Γ, and let ϕ ∶ Γ → ∆ = GΓ be the canonical quotient map. Let Λ be a profinite subgraph ′ of ∆; if there is a profinite subgraph Λ of Γ such that ϕSΛ is a monomorphim of graphs with ′ ′ ′ ϕSΛ (Λ ) = Λ, we say that Λ is a lifting of Λ. ′ 29 Definition 3.3.4 (G-transversal). Let G be a profinite group that acts on a connected profinite graph Γ, and let ϕ ∶ Γ → ∆ = GΓ be the canonical quotient map. A G-transversal or a transversal of ϕ is a closed subset J of Γ such that ϕSJ ∶ J → ∆ is a homeomorphism. Associated with this transversal there is a continuous G-section or section of ϕ, j ∶ ∆ → Γ, i.e., a continuous mapping such that ϕj = id∆ and j(∆) = J.

Note that, in general, J is not a graph.

Deﬁnition 3.3.5 (0-transversal). We say that a transversal J is a 0-transversal if d0(m) ∈ J, for each m ∈ J; in this case we refer to j as a 0-section.

Note that if j is a 0-section, then jd0 = d0j.

Definition 3.3.6 (Fundamental 0-transversal). If the quotient graph ∆ = GΓ 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 ∆, T = j(T ) is a lifting of T , i.e., the restriction of j to T is a morphism of graphs (remember that a lifting is a monomorphism when restricted to T ′).

The next example shows that given the quotient map π ∶ X → GX of a proﬁnite group G acting on a proﬁnite G-space X does not always have a continuous section. This fact is used in Example 3.3.8 below, which shows that G-transversals do not exist in general.

Example 3.3.7 ([2], Example 5.6.8). We construct a profinite G-space X such that the quotient map π ∶ X → GX does not have a continuous section. Let K = {0, 1, −1} be the field of integers modulo 3 with the discrete topology, and let G = {1, −1} be the multiplicative group of K. Let I be an indexing set, and consider the direct product X = M K I of copies of K indexed by I. Then X is a profinite space on which G operates continuously in a natural way. Let π ∶ X → GX be the canonical quotient map. We shall prove that π admits a continuous section if and only if I is countable. If I is countable, by [2], Lemma 5.6.7, page 186, as X is second countable, it admits a continuous section. Conversely, assume that σ ∶ GX → X is a continuous section of π and let Z = Im(σ). Hence Z is a compact subset of X because the section σ is continuous and we have that 0 ∈ Z because the action of G on X fixes 0. The image of a non zero element can be either 1 or −1, it is, either 1 ∈ Z or −1 ∈ Z (not both, because if it happens, Z would have the same generator set as X, so X = Z and the section would be an inverse map). Let J be a finite subset of I and let

30 u = (ui) ∈ X be such that ui = 0 for i ∉ J. Deﬁne

B(J, u) = {x ∈ M K S xj = uj, ∀j ∈ J}. I

Then the subsets of X of the form B(J, u) are clopen and constitute a base for the topology of X. For i ∈ I, write ei for the element of X which has entry 1 at position i and entry 0 elsewhere. Define εi ∈ {1, −1} to be such that εiei ∉ Z. Since Z is closed, for each i ∈ I there exists a finite subset Ji of I such that i ∈ Ji and B(Ji, εiei) ∩ Z = ∅. Consider now any two distinct indices i, j ∈ I. We claim that either i ∈ Jj or j ∈ Ji (or both). To see this, set x = εiei − εjej. Assume that i ∉ Jj and j ∉ Ji. Then, x ∉ Z (since j ∉ Ji implies x ∈ B(Ji, εiei)); similarly, −x ∉ Z (since i ∉ Jj implies x ∈ B(Jj, εjej)). This is a contradiction, and so the claim is proved. Next we show that I is countable. Let N be a countably infinite subset of I and set P = ⋃i∈N Ji. If I were uncountable, there would be some j ∈ I − P , since P is countable. Then, by construction, j ∉ Ji, for any i ∈ N. Therefore, i ∈ Jj by the preceding paragraph. In particular, N ⊆ Jj, contradicting the finiteness of Jj.

In contrast with the situation for abstract groups that act on abstract graphs (cf. [3], Proposition I.14), a general subtree of a C-simply connected proﬁnite subgraph of ∆ need not have a lifting to an isomorphic proﬁnite subgraph of Γ, as the following example shows:

Example 3.3.8 (Quotient G-graph with no lifting of trees or simply connected subgraphs, [1], Example 3.4.2). Let X be a proﬁnite space on which a pro-C group G acts continuously in such a way that the canonical epimorphism ϕ ∶ X → GX does not admit a continuous section (see Example 3.3.7). 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 )S x ∈ X};

• d0(x, P ) = P ;

• d1(x, P ) = x.

We term C the cone of X, represented by the following diagram, where i ∈ I, not necessarily a countable set.

31 xi1 xi9 xi2 . .. xi8 P xi3

xi7 xi4

xi6 xi5

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 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(Xi,P ); ←Ð therefore C is the inverse limit of ﬁnite trees. The quotient graph of C under the action of G is the cone of GX: GC = C(GX,P ).

In particular, GC is a C-simply connected graph (see Proposition 3.2.10) 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(GX,P ) → C(X,P ) such that ϕψ = idC(GX,P ).

The following result proves the existence of 0-sections for Galois coverings:

Lemma 3.3.9 ([1], Lemma 3.4.3). Let ζ ∶ Γ → ∆ be a Galois C-covering of a proﬁnite graph ∆ ′ with associated group G = G(ΓS∆). Assume that Λ is a lifting of a proﬁnite subgraph Λ of ∆. ′ Then there exists a 0-transversal J ⊆ Γ of ζ such that Λ ⊆ J.

Proof. Note that GV (Γ) = V (∆). Since V (Γ) is a proﬁnite space and G acts on it freely, by [2], Lemma 5.6.5, page 185, there is a continuous section

jV ∶ V (∆) → V (Γ)

′ of π = ζSV (Γ) such that, as Y = {d0(m)S m ∈ Λ } ⊆c V (Γ) and πSY is injective, then j can be ′ chosen such that Y is a subset of jV (V (∆)). Thus, let {d0(m)S m ∈ Λ } ⊆ jV (V (∆)). Deﬁne

−1 J = d0 (jV (V (Γ))).

′ Note that Λ ⊆ J and clearly d0(J) ⊆ J. It remains to prove that the restriction ζSJ ∶ J → ∆ of ζ to J is a homeomorphism of topological spaces (cf. Deﬁnition 3.3.4). Since J is

32 compact, it suffices to prove that ζSJ is a bijection. Let m ∈ ∆, and let m˜ ∈ Γ be such that ζ(m˜ ) = m. Put v = jV d0(m). Since ζd0(m˜ ) = ζ(v) = d0(m), (by q-morphism definition, ζ(dj(m)) = dj(ζ(m)), cf. Definition 2.2.7) there exists some g ∈ G with gd0(m˜ ) = v. Put ′ ′ ′ ′ m = gm˜ . Then ζ(m ) = m, and d0(m ) = v, i.e., m ∈ J. So ζSJ 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 ζSJ is also an injection. ˜ Definition 3.3.10. Let ζ ∶ Γ → Γ be the universal Galois C-covering of a profinite graph ˜ Γ. By Lemma 3.3.9 there exists a continuous 0-section j ∶ Γ → Γ of ζ. Let J = 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).

˜ C If m˜ ∈ Γ, deﬁne κ(m˜ ) to be the unique element of π1 (Γ) such that

κ(m˜ )(jζ(m˜ )) = m.˜ (3.1)

For an arbitrary m ∈ Γ one has ζd1j(m) = d1ζj(m) = d1(m) = ζjd1(m) (by q-morphism definition, ζ(dj(m)) = dj(ζ(m)), see Definition 2.2.7). We define χ(m) to be the unique C element of π1 (Γ) such that χ(m)(jd1(m)) = d1j(m). (3.2)

Lemma 3.3.11 ([1], Lemma 3.4.4). The following properties hold for the functions κ = κj and χ = χj deﬁned above.

(a) κd1j(m) = χ(m), ∀m ∈ Γ;

˜ C (b) κ(hm˜ ) = hκ(m˜ ), ∀m˜ ∈ Γ, h ∈ π1 (Γ); ˜ (c) κ(m˜ ) = κ(d0(m˜ )), ∀m˜ ∈ Γ;

(d) κj(m) = 1, ∀m ∈ Γ;

(e) χ(v) = 1, ∀v ∈ V (Γ); ˜ (f) κ(m˜ )(χζ(m˜ )) = κd1(m˜ ), ∀m˜ ∈ Γ;

(g) The maps κ and χ are continuous.

Proof. See [1], Lemma 3.4.4, page 80.

33 One can sharpen Lemma 3.3.9 when the quotient graph ∆ = GΓ is finite. Lemma 3.3.12 ([1], Lemma 3.4.5). Let a profinite group G act on a connected profinite graph Γ and let ϕ ∶ Γ → ∆ = GΓ be the corresponding projection. (a) Let T be a finite subtree of the graph ∆ and let T ′ be a finite subtree of Γ that ϕ sends injectively into T . Then T lifts to a subtree of Γ containing T ′;

(b) Assume further that the action of G on Γ is free and that the quotient graph ∆ = GΓ is ﬁnite. Let m0 ∈ Γ. Then there exists a fundamental 0-transversal J in Γ containing m0.

′ Proof. (a) Let L be the set of finite subtrees of Γ containing T which are sent injectively into ′ T by means of ϕ. Let T0 be a maximal element of L with respect to inclusion, and let T0 be its image in T . Suppose that T0 ≠ T . Since T is finite and connected, there exists an edge e of T not belonging to T0 such that one of the vertices of e is in T0, say d0(e) ∈ V (T0); ′ ′ then d1(e) ∉ V (t0). Let v be a vertex of T0 whose image in T is d0(e). Let e” ∈ Γ with ′ ϕ(e”) = e. Since v and d0(e”) are in the same G-orbit, there exists some g ∈ G with ′ ′ ′ ′ ′ gd0(e”) = v . Define e = ge”. Then d0(e ) = d0(ge”) = gd0(e”) = v and ϕ(e ) = e. Since ′ ′ ′ ′ ′ T0 ∪{e , d1(e )} ∈ L, this contradict the maximality of T0. Thus ϕ(T0) = T0 = T , as desired.

(b) Since ∆ 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 a lifting T of T such ′ −1 ′ ′ that d0(m0) ∈ V (T ). Define J = d0 (V (T )). Note that m0 ∈ J and T ⊆ J. Then arguing as in the proof of Lemma 3.3.9, we see that J is a 0-transversal, and since T ′ is a lifting of a maximal tree of ∆, J is a fundamental 0-transversal. Equivalently, one can describe J ′ ′ ′ ′ more explicitly: for each edge e ∈ ∆ − T , choose e ∈ Γ 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 Γ); then J consists of T ′ together with all the chosen edges e′.

3.4 Existence of universal Galois coverings and the Nielsen- Schreier Theorem for pro-C groups

In this section we show the existence of universal Galois coverings in Construction 3.4.1 and Theorem 3.4.3. We finish the section with an original proof of the Nielsen-Schreier theorem for free profinite groups on a finite space using everything we have stated so far.

Construction 3.4.1 ([1], Construction 3.5.1). Let Γ be a ﬁnite connected graph and let T be a connected 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.

34 Consider the element ∗ = ω(T ) as a distinguished point of X. Let F = 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));

• 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). Clearly this is a free action (because F is a free group) 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 3.4.2 ([1], Lemma 3.5.2). The Galois covering υ ∶ ΥC(Γ,T ) → Γ is connected. Proof. See [1], Lemma 3.5.2, page 83.

Theorem 3.4.3 ([1], Theorem 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 by Proposition 3.2.10). Then one has the following properties:

(a) The Galois C-covering υ ∶ ΥC(Γ,T ) → Γ of Construction 3.4.1 is universal.

(b) Let (X, ∗) = (Γ~T, ∗); then C π1 (Γ) = FC(X, ∗)

is a free pro-C group of ﬁnite rank SΓS − ST S;

Proof. See [1], Proposition 3.5.3, page 84. ˜ Corollary 3.4.4 ([1], 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 ˜ C j ∶ Γ → Γ of ζ lifting T , and let χj ∶ Γ → π1 (Γ) be the corresponding map (see the deﬁnition of χ). Then the pointed space (χj(Γ), 1) with distinguished point 1 is a basis for the free pro-C C group π1 (Γ).

35 Proof. See [1], Corollary 3.5.4, page 86.

Example 3.4.5 ([1], Exercise 3.3.4(a)). Let C be a pseudovariety of ﬁnite groups and L(0) be the loop

0 0

Take a maximal subtree T = {0} of L(0) and ω ∶ L(0) → L(0)~T = L(0). The universal Galois covering of L(0), by Construction 3.4.1 and Theorem 3.4.3, is

ΥC(L(0),T ) = FC(L(0)~T, ω(T )) × L(0) = FC(L(0), {0}) × L(0),

Also by Theorem 3.4.3, rank(F ) = SL(0)S − S{0}S = 2 − 1 = 1, thus F = ZCˆ (where F = FC(L(0), {0})) and

FC(L(0), {0}) × L(0) = ZCˆ × {0}

= Γ(ZCˆ, {1}).

Thereafter ΥC(L(0),T ) = Γ(ZCˆ, 1) is deﬁned by

V (ΥC(L(0), {0})) = V (Γ(ZCˆ, 1))

= {(g, 1)S g ∈ ZCˆ} = ZCˆ;

• d0(r, m) = (r, d0(m));

• d1(r, m) = (rω(m), d1(m));

(r ∈ F , m ∈ L(0)), with an action of F = ZCˆ on the graph ΥC(L(0), {0}) = Γ(ZCˆ, {1}) ′ ′ ′ by r (r, m) = (r r, m), (r, r ∈ F, m ∈ L(0)). Therefore, ZCˆΓ(ZCˆ, {1}) = L(0) and C π1 (L(0)) ≅ ZCˆ.

Example 3.4.6 ([1], Exercise 3.3.4(b)). Let C be a pseudovariety of ﬁnite groups and n be a natural number. Consider the ﬁnite graph L(n)

0 1 2 3 ... n − 1 n 0 1 2 n − 1 n

Take a maximal subtree T of L(n),

36 0 1 2 3 ... n − 1 n 0 1 2 n − 1 and ω ∶ L(n) → L(n)~T = L(0), ω(T ) = {0}. The universal Galois covering of L(0), by Construction 3.4.1 and Theorem 3.4.3, is

ΥC(L(n),L(0)) = ΥC(L(0), {0}) = FC(L(0), {0}) × L(n),

By Example 3.4.5, FC(L(0), {0}) ≅ ZCˆ. Therefore, ΥC(L(n),L(0)) = Γ(ZCˆ, {1}), C C ZCˆΓ(ZCˆ, {1}) = L(0) and π1 (L(n)) ≅ π1 (L(0)) ≅ ZCˆ.

The next results are needed for the proof of the Nielsen-Schreier Theorem for proﬁnite groups:

Theorem 3.4.7 ([1], Proposition 3.6.1). Let ζ ∶ Γ̃ → Γ be a universal Galois covering of a C connected proﬁnite graph Γ and let H be a closed subgroup of π1 (Γ).

(a) The canonical epimorphism ξ ∶ Γ̃ → HΓ̃ is a universal Galois covering

C (b) π1 (HΓ̃) = H.

Proof. See [1], Proposition 3.6.1, page 89.

Theorem 3.4.8 ([1], Proposition 3.7.4). Let Γ be a connected proﬁnite graph having a spanning C-simply connected proﬁnite subgraph T . The following properties hold:

(a) The Galois C-covering υ ∶ Υ(Γ,T ) → T constructed in 3.4.1 is universal.

C (b) The fundamental group π1 (Γ) is a free pro-C group on the pointed proﬁnite space (X, ∗) = (Γ~T, ∗).

Proof. See [1], Theorem 3.7.4, page 94.

Theorem 3.4.9 ([1], Proposition 3.8.1). The Cayley graph Γ(F (X, ∗),X) of a free proﬁnite group on a pointed proﬁnite space (X, ∗) with respect to X is simply connected. In fact, Γ(F (X, ∗),X) is the universal Galois covering space of the bouquet of loops B = B(X, ∗) and π1(B) = F (X, ∗).

Proof. See [1], Proposition 3.8.1, page 95.

We finish this chapter with an original proof of the Nielsen-Schreier Theorem for profinite groups on a finite space. This Theorem is proved in [2], from the article of Binz, Neukirch and Wenzel [1971] and uses the abstract version. Ribes and Steinberg (2010) gave a new proof without using the abstract version, through wreath products, but it is not so simple as the one presented in this thesis. The approach is entirely different as well.

37 Theorem 3.4.10 (Profinite version of the Nielsen-Schreier theorem on a finite space). Let F be a free profinite group on a finite space X and H be an open subgroup of F . Then H is a free profinite group.

Proof. Let Γ(F,X) be the Cayley graph of F . As F acts freely on Γ(F,X) (cf. Example 2.3.12), the Cayley graph Γ(F,X) is a Galois covering of F Γ(F,X) and by Theorem 3.4.9, Γ(F,X) is the universal Galois covering of F Γ(F,X) = B(X, ∗) and π(Γ(F,X)) = F . Similarly, as H is a closed subgroup of F with ﬁnite index and π1(Γ(F,X)) = F , by Theorem 3.4.7 the canonical epimorphism ξ ∶ Γ(F,X) → H Γ(F,X) is a universal Galois covering and π1(H Γ(F,X)) = H. Hence we have the following diagram:

π1(H Γ(F,X))=H Γ(F,X) / H Γ(F,X)

π1(Γ(F,X))=F

F Γ(F,X)

However, the vertices of Γ(F,X) can be mapped homeomorphically to the elements of the free profinite group F as stated in Example 2.2.22. Thus, as [F ∶ H] is finite, H Γ(F,X) will have finitely many vertices and as X is finite, E(Γ(F,X)) is finite, which implies that H Γ(F,X) is finite. Hence, by Theorem 3.4.3,

H = π1(H Γ(F,X)) = F ((H Γ(F,X))~T, ∗), where T is the maximal subtree of H Γ(F,X). Therefore H is free, as desired.

38 Chapter 4

Graphs of pro-C groups

In this chapter we deﬁne a sheaf, free product of a sheaf, a graph, the fundamental group and the standard graph of pro-C groups. We also show that we can see a free product of a sheaf of pro-C groups as a fundamental group of a graph of pro-C groups and use it to prove the Kurosh Subgroup Theorem in the last section, the main result of this thesis.

4.1 Free pro-C product of a sheaf of pro-C groups

In this section we deﬁne a sheaf of pro-C groups and the free pro-C product of a sheaf.

Definition 4.1.1 (Sheaf). 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 , the ﬁber G(t) = π 1(t) over t is a pro-C group whose topology is induced by the topology of G as the subspace topology;

(b) If we deﬁne 2 G = {(g, h) ∈ G × G S π(g) = π(h)},

2 −1 then the map µ ∶ G → G given by µG(g, h) = gh is continuous.

Example 4.1.2 ([1], Example 5.1.1(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 union 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). For the second condition, 2 G = {(g, h) ∈ G × G S g, h ∈ Gi} and we can write G = "i∈T Gi × Gi, so each Gi × Gi is clopen. Take U an open subset of Gi. Then the map µ restricted to it is continuous because Gi is a −1 2 topological group and µ (U) is open in Gi × Gi. Hence, as Gi × Gi is open in G , we have that − µ 1(U) is open in G2, as desired.

39 Example 4.1.3 ([1], Example 5.1.1(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.

′ ′ ′ ′ Deﬁnition 4.1.4 (Morphism of sheaves). A morphism α = (α, α ) ∶ (G, π, T ) → (G , π ,T ) of ′ ′ ′ sheaves of pro-C groups consists of a pair of 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 .

Example 4.1.5. Let (G, π, T ) be a sheaf of pro-C groups and let H be a pro-C group. We may think of H as the ﬁber of a sheaf over a singleton space.

α G / H

π π ′ α / T ′ 1

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.

Deﬁnition 4.1.6 (Monomorphism). A morphism α is said to be a monomorphism if α and α′ are injective.

The deﬁnition of epimorphism is analogue.

′ Deﬁnition 4.1.7 (Subsheaf). The image of a monomorphism G → G is called a subsheaf of the ′ sheaf G .

′ Example 4.1.8. If (G, π, T ) is a sheaf and T is a closed subspace of T , then the triple

−1 ′ ′ (π (T ), πSπ 1(T ),T ) − ′ 40 is a subsheaf of (G, π, T ). Note that this is well defined, because a closed subspace of a profinite space is profinite and the map is just the restriction.

Deﬁnition 4.1.9 (Free pro-C product of a sheaf). A free pro-C product of the sheaf (G, π, T ) is deﬁned to be a pro-C group G together with a morphism ω ∶ G → G (see Example 4.1.5), that respects the following universal property: for every morphism β of the sheaf G into a pro-C group H there exists a unique continuous homomorphim β ∶ G → H such that the following diagram:

ω G / G

β β

H commutes, that is: βω = β. Remark 4.1.10. 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. Indeed, let H = lim Hi and Y a ←Ði∈I topological space. By using the universal property of the inverse limit (cf. Deﬁnition 2.1.4), we have the following

ω G / G

β β " ψ H = lim Hi o Y ←Ð

ϕi ψi

| Hi

And we can rewrite it as

ω G / G

βϕi βϕi Hi

Proposition 4.1.11 (Existence and uniqueness of the free pro-C product, [1], 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.

41 Proof. The uniqueness follows 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 (cf. [3], Example 1, I.2) and 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. If N1,N2 ∈ N , then N1 ∩ N2 ∈ N , because the composition map is continuous. 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, that is

KN (L) = lim L~N. N←Ð∈N

Then KN (L) is a pro-C group. Let ı ∶ L → KN (L) be the natural map. Put ω = ıϕ; 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 from the sheaf G to the pro-C group H (cf. Example 4.1.5). We claim that (KN (L), ω) is the free pro-C product of G. To see this we check the corresponding universal property. By Remark 4.1.10, we only need to check the universal property for groups H ∈ C. Hence, let H ∈ C and let β ∶ G → H be a morphism.

KN (L) β ı 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 = βı. Hence

42 β = βω. The uniqueness of β follows from the fact KN (L) = ⟨ω(G)⟩.

The free pro-C product of a sheaf of pro-C groups (G, π, T ) will be denoted by ∐T G.

Example 4.1.12 ([1], 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 deﬁned in Example 4.1.2: G is just the disjoint union of the groups G1, ⋯Gn. The corresponding free pro-C product G = ∐ G coincides with the standard concept of a free pro-C product of the groups Gi (cf. [2], Section 9.1, page 353) and it is usually written

n G = G1 ⋯ Gn = O Gi. i=1

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 homomorphims ϕi ∶ Gi → H (i = 1, ⋯, n), there exists a unique continuous homomorphism ϕ ∶ G → H such that ϕωi = ϕi (i = 1, ⋯, n).

Example 4.1.13 ([1], Example 5.1.3(b)). Let T be a profinite space. Consider the constant sheaf KT (ZCˆ) = (T × ZCˆ, π, T ) (see Example 4.1.3). Then the free pro-C product ∐T KT (ZCˆ) is isomorphic to the free pro-C group FC(T ) on the profinite space T . Indeed, by the definition of the free pro-C product we have the following

ω T × ZCˆ / ∐T KT (ZCˆ)

β β

$ H

Adding the projection map π ∶ T × ZCˆ → T map, we have:

ω T × ZCˆ / ∐T KT (ZCˆ)

π β β

$ TH

Now take the compositions ωı ∶ T → ∐T KT (ZCˆ) and βı ∶ T → H, (where ı ∶ T → T × ZCˆ

43 deﬁned by t ↦ (t, 0) is the inclusion map) that produces the diagram:

β ∐T KT (ZCˆ) / H O ;

ωı βı

Therefore, as they respect the same universal property, the free pro-C product ∐T KT (ZCˆ) is isomorphic to the free pro-C group FC(T ) on the proﬁnite space T , as desired. C Proposition 4.1.14 ([1], Example 5.1.6). Let (G, π, T ) be a sheaf of pro-C groups, G = ∐T G, 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 . Proof. See [1], Proposition 5.1.6, page 142. Definition 4.1.15. 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 S Gt ⊆ U} of T is open in T .

Lemma 4.1.16 ([1], Example 5.2.1). Let G be a proﬁnite group and let {Gt S t ∈ T } be a collection of closed subgroups of G indexed by a proﬁnite space T . The family {Gt S t ∈ T } is continuously indexed by T if and only if the set

E = {(t, g) ∈ T × G S t ∈ T, g ∈ Gt} is a closed subset of T × G. Proof. See [1], Lemma 5.2.1, page 145. Lemma 4.1.17 ([1], Example 5.2.2). Let G be a profinite group that acts continuously on a profinite space T . Then the collection {Gt S t ∈ T } of the G-stabilisers 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)S t ∈ T } of T × T is closed because the space T × T with the product topology is profinite, it follows that

−1 E = {(t, g) ∈ T × G S g ∈ Gt} = α (∆)

44 is closed in T × G by the continuity of the map α. The result now follows from Lemma 4.1.16.

Proposition 4.1.18 ([1], Example 5.2.3). Let G be a proﬁnite group and let

F = {Gr S r ∈ T } be a continuously indexed family of closed subgroups of G, where T is a proﬁnite space. Consider the equivalence relation on the product 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. See [1], Proposition 5.2.3, page 146.

4.2 Graphs of pro-C groups and specialisations

In this section we deﬁne a graph of pro-C groups and specialisations and give some examples.

Definition 4.2.1 (Profinite graph of pro-C groups, [1], Definition 6.1.1). Let Γ be a connected profinite graph with incidence maps d0, d1 ∶ Γ → V (Γ). We can define a profinite graph of pro-C groups over Γ as a sheaf (G, π, Γ) of pro-C groups together with two morphisms of sheaves

(B0, d0), (B1, d1) ∶ (G, π, Γ) → (GV , πSπ 1(V (Γ)),V (Γ)) −

[here GV denotes the restriction subsheaf of G to the space V (Γ), that we term the ’vertex subsheaf of G’. Note that it is indeed a sheaf, because the vertex set V (Γ) is closed in Γ, so it is also a profinite space and the vertex sheaf is well defined (cf. Example 4.1.8)], where the restriction of Bi to GV is the identity map idGV (i = 0, 1); in addition, we assume that the restriction of Bi to each fiber G(m) is an injection (m ∈ Γ), (i = 0, 1).

Deﬁnition 4.2.2. 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. Choose a continuous 0-section j of ζ, and denote by J = j(Γ) the corresponding 0-transversal (that exists by Lemma 3.3.9). Associated with j there is a continuous function

C χ ∶ Γ → π1 (Γ)

45 from Γ into the fundamental group of Γ deﬁned by

χ(m)(jd1(m)) = d1j(m)

(See Equation 3.2 on Section 3.3)

Deﬁnition 4.2.3 (J-specialisation). Given a pro-C group H, deﬁne a J-specialisation 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) = βB0(x) = (β χ(m))(βB1(x))(β χ(m)) (4.1) for all x ∈ G, where m = π(x). Note that the deﬁnition of the map χ depends uniquely on the 0-section j and the corresponding G-transversal J (cf. Deﬁnition 3.3.10). The following diagram shows the compositions:

β G H β βS V ′ C (4.2) Bi G π1 (Γ) π χ

GV Γ

Example 4.2.4 ([1], 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 ∈ Γ. Then we may think of a J-specialisation ′ (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 (4.1) are automatic in this case. Indeed, (4.1) gives us that:

′ ′ −1 β(G(m)) = βB0(G(m)) = (β χ(m))(βB1(G(m)))(β χ(m))

So, if m ∈ E(Γ), we have that:

β(G(e)) = βB0(G(e))

β(1) = βB0(1).

46 By diagram 4.2, β G / H >

B0 βS V G GV so this diagram commutes for every e ∈ E(Γ). If v ∈ V (Γ), B0(G(v)) = idG(V ) = G(v), then

β(G(v)) = βB0(G(v)) β(G(v)) = β(G(v)). and we have nothing to show. Thus, for B1 and e ∈ E(Γ), we have:

′ ′ −1 β(G(e)) = (β χπ(G(e)))(βB1(G(e)))(β χπ(G(e))) ′ ′ −1 β(1) = (β χπ(1))(βB1(1))(β χπ(1))

By diagram 4.2, β G H β βS V ′ C Bi G π1 (Γ) π χ

GV Γ and the condition of Equation (4.1) is automatically satisﬁed.

C ˜ Example 4.2.5 ([1], Example 6.1.2(b)). If Γ is C-simply connected, then π1 = 1 and Γ = Γ = J. Then we can refer to a ’specialisation’ rather than ’J-specialisation’: it is just a morphism β ∶ G → H such that β(x) = βB0(x) = βB1(x) for all x ∈ G. It happens for example if Γ is a ﬁnite tree or, more generally, an inverse limit of ﬁnite trees (see Proposition 3.2.10).

4.3 The fundamental group of a graph of pro-C groups

In this section we deﬁne the fundamental group of a graph of pro-C groups through a universal property and ﬁnish the section with Example 4.3.6, which shows the free pro-C product of a sheaf as the fundamental group of a graph of pro-C groups.

Deﬁnition 4.3.1 (The fundamental group of a graph of pro-C groups). Choose a continuous 0- ˜ section j of the universal Galois C-covering ζ ∶ Γ → Γ and denote by J = j(Γ) the corresponding

47 0-transversal (see Lemma 3.3.9). We deﬁne a fundamental pro-C group of the graph of groups C (G, π, Γ) with respect to the 0-transversal J to be a pro-C group Π1 (G, Γ), together with a ′ C J-specialisation (ν, ν ) of (G, π, Γ) in Π1 (G, Γ) satisfying the following universal property:

C Π1 (G, Γ)

ν ν′

C G δ π1 (Γ)

β β′

′ whenever H is a pro-C group and (β, β ) a J-specialisation of (G, π, Γ) in H, there exists a unique continuous homomorphism

C δ ∶ Π1 (G, Γ) → H

′ ′ ′ such that δν = β and δν = β . We refer to (ν, ν ) as a universal J-specialisation of (G, π, Γ). Remark 4.3.2. 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 (cf. Remark 4.1.10).

Proposition 4.3.3 (Existence of fundamental groups, [1], Proposition 6.2.1). 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

C (a) There exists a fundamental pro-C group Π1 (G, Γ) of (G, π, Γ) with a universal J- ′ specialisation (ν, ν );

C (b) (uniqueness for a ﬁxed J) The fundamental pro-C group Π1 (G, 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-specialisation of (G, π, Γ) in Π, then there exists a unique C ′ ′ continuous isomorphism ξ ∶ Π1 (G, Γ) → Π such that ξν = µ and ξν = µ ;

C ′ C (c) Π1 (G, Γ) is topologically generated by {ν(G(v)) S v ∈ V = V (Γ)} and ν (π1 (Γ)).

Proof. Part (b) follows from the universal property. Consider the free pro-C product of the vertex sheaf (see Section 4.1), W = O G(v) = O GV . v∈V We can do an embedding of each vertex group each vertex group G(v) in W under the natural morphism

ω ∶ Gv → W

48 (cf. Proposition 4.1.14(c)). Deﬁne

C C Π1 (G, Γ) = (W π1 (Γ))~N

C where π1 (Γ) denotes the free pro-C product of pro-C groups, and where N is the topological C closure of the normal subgroup of the group W π1 (Γ) generated by the set

−1 −1 {B0(x) (χπ(x))B1(x)(χπ(x)) S x ∈ G}.

C Deﬁne ν ∶ G → Π1 (G, Γ) to be the composition of the natural continuous maps

B0 ω ı C α C C G / GV / W / W π1 (Γ) / Π1 (G, Γ) = (W π1 (Γ))~N,

′ C C where ı is the inclusion and α is the quotient map and ν ∶ π1 (Γ) → Π1 (G, Γ) is the composition of the natural continuous homomorphisms

ı α πC Γ / W πC Γ / ΠC , Γ W πC Γ N 1 ( ) ′ 1 ( ) 1 (G ) = ( 1 ( ))~

′ ′ where ı is the inclusion. We want to verify that (ν, ν ) is indeed a J-specialisation of the graph C of pro-C groups (G, π, Γ) in Π1 (G, Γ). C By the deﬁnition of N, it is the closure of the normal subgroup of the group W π1 (Γ) generated by the set −1 −1 {B0(x) (χπ(x))B1(x)(χπ(x)) S x ∈ G},

−1 C and it means that B0(x) = (χπ(x))B1(x)(χπ(x)) in Π1 (G, Γ). Therefore, as ν = αıωB0 and ′ ′ ν = αı , we have that

′ ′ −1 νB0(x) = (ν χπ(x))(νB1(x))(ν χπ(x)) and as J is a 0-transversal, ν(x) = νB0(x) for all x ∈ G as desired. C ′ It follows that the pro-C group Π1 (G, 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.

C The special case when π1 (Γ) = 1 is very important and will be used in the proof of the C Kurosh Subgroup Theorem (KST). To say that π1 (Γ) = 1 is the same as consider Γ a C-simply ˜ ˜ connected proﬁnite graph. Then there is only one 0-transversal J of ζ ∶ Γ → Γ, that is J = Γ = Γ. C In this case we can refer just to Π1 (G, Γ) (without considering the transversal) because there is only one choice for J. The following corollary is an easy consequence of Proposition 4.3.3(c).

Corollary 4.3.4 ([1], Corollary 6.2.2). Let (G, π, Γ) be a graph of pro-C groups over a C-simply C connected proﬁnite graph Γ. Then Π1 (G, Γ) is generated as a topological group by the images ν(G(v)) (v ∈ V (Γ)) of the vertex groups.

49 Example 4.3.5 ([1], Example 6.2.3(a)). Assume that the graph of pro-C groups (G, π, Γ) satisﬁes G(e) = 1, for all edges e ∈ Γ. ′ As in Example 4.2.4, we can think of a J-specialisation (β, β ) in a pro-C group H as a morphism β from G to H, that is, J plays no role. In this case the fundamental pro-C group of this graph of pro-C groups is just the free pro-C product

C C Π1 (G, Γ) = O G(v) π1 (Γ), v∈V (Γ)

C because the subgroup N of W π1 (Γ) is trivial. We may take the universal specialisation to be ′ (ν, ν ) as constructed in Proposition 4.3.3, where ν is the composition

≅ C G ∐ G ∐v∈V (Γ) G(v) ∐v∈V (Γ) G(v) π1 (Γ).

[the homomorphism ∐ G → ∐v∈V (Γ) G(v) is induced by B0 ∶ G → GV , and in this case it is an isomorphism], while ν′ is the natural inclusion

C C π1 (Γ) ∐v∈V (Γ) G(v) π1 (Γ).

Example 4.3.6 ([1], Example 6.2.3(b)). Let X be a profinite space and G = ∐X Gx be a free pro-C product, where the set {Gx S x ∈ X} is continuously indexed by the profinite space X (cf. Definition 4.1.15). Then G can be viewed as the fundamental group of a graph of groups. To see this, first take a single point space {ω} disjoint from X and construct a profinite graph T = T (X) as follows:

• V (T ) = X ⊍ {ω};

• E(T ) = {ω} × X = {(ω, x)S x ∈ X};

• d0(ω, x) = ω;

• d1(ω, x) = x (x ∈ X), where V (Γ) and T = V (T ) ⊍ E(T ) are endowed with the disjoint union topology and E(Γ) with the product topology. Observe that T is an inverse limit of ﬁnite trees T (Xi). 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 . ⎩⎪ = ∈ { } ⊍ ( )

From Lemma 4.1.16 we have 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 B0 and B1; they are the identity maps on vertex groups and trivial homomorphisms otherwise.

50 Gx1

Gx2 1 1 Gx3 1 1 1 . .. Gx

C Since T is an inverse limit of ﬁnite trees, by Proposition 3.2.10(b), π1 (T ) = 1; hence, as a particular instance of Example 4.3.5, we have

C G = O GX = Π1 (G,T ). X

Example 4.3.7. Let (G, π, Γ) be a graph of pro-C groups over a ﬁnite graph Γ and T a maximal subtree of Γ. Using the idea of abstract Bass-Serre theory, we can think of the subset C {te S e ∈ E(Γ) − E(T )} as a basis for the free pro-C group π1 (Γ). Then, by the construction of Proposition 4.3.3, we have that

C ⎛⎛ ⎞ ⎞ Π = Π1 (G, Γ) = O G(v) FC ~N, ⎝⎝v∈V (Γ) ⎠ ⎠ where FC is the free pro-C group with basis {te S e ∈ E(Γ)} and N is the smallest closed normal subgroup of ∐v∈V (Γ) G(v) FC containing the set

−1 −1 {te S e ∈ E(T )} ∪ {B0(x) teB1(x)te S x ∈ G(e), e ∈ E(Γ)}.

For each v ∈ V (Γ), νv ∶ G(v) → Π is the natural continuous homomorphism x ↦ xN.

4.4 The standard graph of a graph of pro-C groups

In this section we deﬁne the standard graph of a graph of pro-C groups, which can be seen as the C-universal covering of a graph of pro-C groups with respect to a pro-C group H, by Theorem 4.4.5.

Definition 4.4.1 (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, j ∶ Γ → Γ be a continuous 0-section of the universal ˜ Galois C-covering ζ ∶ Γ → Γ of Γ (that exists by Lemma 3.3.9), and let J = j(Γ) be the ′ corresponding 0-transversal (see Section 4.1). Let (γ, γ ) be a J-specialisation of (G, π, Γ) in a pro-C group H (cf. Definition 4.2.3). Then we can define a profinite graph

C S (G, Γ,H)

51 which is canonically associated to the graph of groups (G, π, Γ) and H. The following construction is done in a way that the quotient of the action of the fundamental group on this graph of pro-C groups is Γ. For m ∈ Γ, deﬁne H(m) = γ(G(m)). Then the collection

{H(m)S m ∈ Γ} of closed subgroups of H is continuously indexed by Γ (cf. Deﬁnition 4.1.15). Indeed, if U is − an open subset of H, then γ 1(H − U) is a closed and therefore a compact subset of G; since

−1 Γ − {m ∈ Γ S H(m) ⊆ U} = π(γ (H − U)),

− it follows that Γ(U) = {m ∈ Γ S H(m) ⊆ U} is open in Γ, because π(γ 1(H − U)) is compact (by the continuity of π), and therefore closed. C As a topological space, S (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).

C So, as a set, S (G, Γ,H) is the disjoint union

C S (G, Γ,H) = # H~H(m). m∈Γ

C By Proposition 4.1.18, S (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

C p ∶ S (G, Γ,H) → Γ,

− ′ such that p 1(m) = H~H(m) and p = pα.

p Γ H / Γ × ′ 9 α p C S (G, Γ,H)

C To make S (G, Γ,H) into a proﬁnite graph we deﬁne the subspace of vertices of C S (G, Γ,H) by C −1 V (S (G, Γ,H)) = p (V (Γ)). The incidence maps C C di ∶ S (G, Γ,H) → V (S (G, Γ,H))

52 (i = 0, 1) are deﬁned as follows:

d0(hH(m)) = hH(d0(m)) (4.3)

′ d1(hH(m)) = h(γ χ(m))H(d1(m)), (4.4)

C (h ∈ H, m ∈ Γ), where χ ∶ Γ → π1 (Γ) is the continuous map considered in Equation 3.2, Section ′ 3.3. Note that (γ χ(m)) ∈ H, as the following composition diagram shows:

χ C γ Γ π1 (Γ) ′ H.

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. Indeed, by the definition of H(m) we have that H(m) ≤ H(d0(m)) if and only if

γ(G(m)) ≤ γ(G(d0(m))).

But it happens because G(m) ↪ G(d0(m)) is an embedding. Let y ∈ 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 Equation (4.1), this is true since y = γ(x), for some x ∈ G(m), and B1(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 diagram

di Γ × H / V (Γ) × H

α αV SC , Γ,H / V SC , Γ,H (G ) di ( (G ))

′ (i = 0, 1), where d0(m, h) = (d0(m), h) and d1(m, h) = (d1(m), (γ χ(m)h)(m ∈ Γ, h ∈ H). C C Since the topologies of S (G, Γ,H) and V (S (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

53 ′ obvious, and for d1 it follows from the continuity of γ , χ and of the multiplication in H. This C completes the definition of S (G, Γ,H). C The profinite graph S (G, Γ,H) 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.

C We can deﬁne a natural continuous action of H on the graph S (G, Γ,H) given by

′ ′ h(h H(m)) = hh H(m)

′ (h, h ∈ H, m ∈ Γ).

Definition 4.4.2 (C-standard graph). For a technical reason we will assume that C is extension C closed (cf. Definition 2.1.14), so the definition of Π1 (G, Γ) is independent of the G-transversal J (cf. [1], Theorem 6.2.4, page 189). If

C H = Π = Π1 (G, Γ)

′ ′ and (γ, γ ) = (ν, ν ) is the universal J-specialisation of the graph of pro-C groups (G, π, Γ) in C C C Π = Π1 (G, Γ), we use instead the notation S = S (G, Γ) rather than S (G, Γ, Π), and refer to

C S = S (G, Γ) = # Π~Π(m) m∈Γ

(Π(m) = ν(G(m)), m ∈ Γ) as the C-standard graph (or C-universal graph) of the graph of pro-C groups (G, π, Γ).

Example 4.4.3 ([1], Example 6.3.1). Let Γ be a ﬁnite graph and (G, Γ) a graph of pro-C groups C over Γ. Using the notation of Example 4.3.7, the C-standard graph S = S (G, Γ) has vertices and edges

V (S) = ⋃v∈V (Γ) Π~Π(v) and E(S) = ⋃e∈E(Γ) Π~Π(e) and incidence maps

d0(gΠ(e)) = gΠ(d0(e)), d1(gΠ(e)) = gteΠ(d1(e)),

(g ∈ Π, e ∈ E(Γ)). Before stating Lemma 4.4.5, we have a technical result:

Lemma 4.4.4. Let G be a profinite group, X a profinite G-space and σ ∶ G × X → X be an ′ −1 action of G on X with a ∈ G, s ∈ X such that s = as = σ(a, s). Then Gs = aGsa . ′ ′ ′ ′ Proof. Take an element g ∈ Gs . So g stabilises s and we have that gs = s . By definition, ′ ′ s = as, so g(as) = as. The associativity of the action gives to us that g(as) = (ga)s, so − − − (ga)s = as. Multiplying both sides by a 1 we have that (a 1ga)s = s. Therefore a 1ga stabilises s, as desired.

54 C Lemma 4.4.5 ([1], Example 6.3.2). Assume that C is extension-closed. Let Π = Π1 (G, Γ) and C S = S (G, Γ) be as above.

(a) The quotient space ΠS is Γ. Furthermore, the Π-stabiliser 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.

(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. By the natural action, given g ∈ Π, we have that g(g Π(m)) = gg Π(m), and as Π(m) is continuously indexed by Γ, ΠS = Γ. The other statement of (a) follows from Lemma 4.4.4. To prove (b), note that the map σ1 ∶ Γ → Γ × Π given by σ1(m) = (m, 1)(m ∈ Γ) is a continuous section of the natural projection pr1 ∶ Γ × Π → Π. Let σ be the composition of σ1 and the quotient map α ∶ Γ × Π → S, i.e., σ(m) = 1Π(m)(m ∈ Γ).

σ { 1 / Γ × H pr1 Γ

α σ

} S

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 ∈ T , by Equation (4.3). Hence it suffices to show that the injection σST ∶ T → S is a morphism of graphs. This follows from the fact that the function χ used in formula 4.4 can be chosen so that χ(m) = 1 if m ∈ T (see Theorem 3.4.8 and the definition of χ).

Let Σ be a profinite graph, G a profinite group, Γ = GΣ be a finite graph and T a maximal ˜ ˜ subtree of Γ . Construct a fundamental 0-section j ∶ Γ → Γ of the quotient morphism Γ → Γ

55 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) are in the same G-orbit; choose tm ∈ G such that

tmδd1(m) = d1δ(m), (4.5) and note that tm = 1, for m ∈ T . Similarly, for each m ∈ Γ, we can take the map χ (cf. Deﬁnition 3.3.10) and deﬁne a graph of pro-C groups (G, Γ) as follows: for m ∈ Γ, put

G(m) = Gδ(m), the G-stabiliser of the element δ(m) under the action of G. The incidence morphisms B0 and B1 are deﬁned as follows: on each ﬁber G(m), B0 is just the inclusion map

G(m) ↪ G(d0(m)), and B1 is the composition of an inclusion and conjugation by tm:

−1 m G d m G G 1 t G t , G( ) = δ(m) → G( 1( )) = δd1(m) = tm δd1(m) = m δd1(m) m − −1 x ↦ tm xtm, (x ∈ G(m)) by Lemma 4.4.4 and Equation (4.5). Let ′ C γ ∶ π1 (Γ) → G

′ be the unique continuous homomorphism such that γ χ(m) = tm (m ∈ Γ) (see Corollary 3.4.4). On the other hand, the inclusion maps

G(m) = Gδ(m) ↪ G

(m ∈ Γ) determine a morphism γ ∶ G → G.

′ So (γ, γ ) is a J-specialisation of G in G, where J = j(Γ). Hence it induces a continuous homomorphism C ϕ ∶ Π1 (G, Γ) → G,

′ ′ ′ such that ϕν = γ and ϕν = γ , where (ν, ν ) is the universal J-specialisation of (G, π, Γ) in C Π1 (G, 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 4.4.6 ([1], 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 = GΣ is ﬁnite. Construct a graph of pro-C groups (G, Γ) over Γ as above. Then the homomorphism

C ϕ ∶ Π1 (G, Γ) → G

56 defined above is an isomorphism of profinite groups. Moreover, Σ is isomorphic to the standard C graph S (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, we have

′ ′ ′ Ψ d0(gG(m)) = Ψ (gG(d0(m))) = gδ(d0(m)) = d0(gδ(m)) = d0Ψ (gG(m)), 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. C Since Σ is C-simply connected, so is S (G, Γ,G). Therefore by [1], Theorem 6.3.7, the C C homomorphism ϕ ∶ Π1 (G, Γ) → G is an isomorphism, and S (G, Γ,G) is isomorphic to C C S (G, Γ). It follows that Σ and S (G, Γ) are isomorphic.

4.5 The Kurosh Theorem for free pro-C products

In this last section we will prove the pro-C version of the Kurosh subgroup theorem, the main result of this thesis, by seeing the free pro-C product of a sheaf as a fundamental group of a graph of pro-C groups (see Example 4.3.6). Deﬁnition 4.5.1 (Double cosets). If H and K are subgroups and x is an element of a group G, the subset HxK = {hxk S h ∈ H, k ∈ K} is called an (H,K)-double coset. There is a partition of the group into double cosets:

Proposition 4.5.2. Let H and K be subgroups of a group G.

(a) The group G is a union of (H,K)-double cosets;

(b) Two (H,K)-double cosets are either equal or disjoint;

57 (c) The double coset HxK is a union of the right cosets of H and a union of left cosets of K.

Proof. Deﬁne x ∼ y to mean that x = hyk for some h ∈ H and k ∈ K. It is easy to check that ∼ is an equivalence relation on G, the equivalence class containing x being HxK. Thus (a) and (b) follow at once, (c) is clear.

Theorem 4.5.3 (Kurosh, [1], Theorem 7.3.1). Let C be an extension-closed pseudovariety of n ﬁnite groups. Let G = ∐i=1 Gi be a free pro-C product of a ﬁnite number of pro-C groups Gi. If H is an open subgroup of G, then

n −1 H = O O (H ∩ gi,τ Gigi,τ ) F i=1 τ∈HG~Gi

n Proof. By Example 4.3.6 we can see G = ∐i=1 Gi as the fundamental group of a finite graph of pro-C groups (G, Γ). Indeed, let X be the finite set {1, 2, ⋯, n} and Γ be defined as follows:

• V (Γ) = X ⊍ {ω};

• E(Γ) = {ω} × X = {(ω, x)S x ∈ X};

• d0(ω, x) = ω;

• d1(ω, x) = x (x ∈ X), where {ω} is a single point space disjoint from X and V (Γ), Γ = V (Γ) ⊍ E(Γ) are endowed with the disjoint union topology and E(Γ) with the product topology. Hence we deﬁne the graph of pro-C groups by G as the subset of Γ × G consisting of those elements (m, y) ∈ T × G such that ⎧ ⎪ y ∈ Gm, if m ∈ X; ⎨ ⎪ y 1, if m ω E Γ . ⎩⎪ = ∈ { } ⊍ ( )

From Lemma 4.1.16 we have that G is a proﬁnite space, and so (G, π, Γ) is a subsheaf over Γ of the constant sheaf Γ × G. In fact, (G, π, Γ) has the structure of a graph of pro-C groups with obvious morphisms B0 and B1.

58 The following diagram represents this graph of groups:

Gx1

Gx2 1 1 Gx3 1 1 1 . .. Gx

We denote the vertices of Γ by ω, 1, ⋯, n and 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) because it is a morphism of sheaves over a singleton (cf. Example 4.1.5). 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 groups (G, Γ) (cf. Definition 4.4.2). Since Γ is finite and H is open in G (and so a closed subgroup of finite index in G), the quotient graph

′ Γ = HS = # HG~G(m) m∈Γ is ﬁnite. Hence, we have the following diagram:

S δ

′ Γ = GS Γ = HS

Let X = {1G(m)S m ∈ Γ} be a subgraph of S. According to Lemma 4.4.5, X is a subtree of S isomorphic to Γ. Hence X is isomorphic to the subtree

Y = {H1G(m)S m ∈ Γ} of Γ′. Therefore, by Proposition 3.3.12 and Lemma 3.3.9, there exists a fundamental 0-section ′ δ ∶ Γ → S such that

δ(H1G(m)) = 1G(m), (4.6)

C ′ ′ ′ ′ for every m ∈ Γ. By Theorem 4.4.6, H = Π1 (H, Γ ), where H(m ) = Hδ(m ), for m ∈ Γ . ′ ′ Observe that in this case the edge groups of (H, Γ ) are trivial. Indeed, by Equation (4.6), if

59 ′ ′ ′ ′ m ∈ E(Γ ) then H(m ) = Hδ(m ) and δ(m ) = 1G(m). But G(e) = 1 for all e ∈ E(Γ) and δ is a morphism by the deﬁnition of′ the fundamental 0-transversal; so (see Example 4.3.5)

C ′ C ′ H = Π1 (H, Γ ) = O H(v) π1 (Γ ). v∈V (Γ ) ′ Fix m ∈ Γ; then δ(HG~G(m)) = {gm,τ G(m)S τ ∈ Im}, and {gm,τ S τ ∈ Im} is a system of representatives of the double cosets HG~G(m); furthermore, by our choice of δ, one has that

1 ∈ {gm,τ S τ ∈ Im}.

′ ′ ′ If m ∈ HG~G(m) ⊆ Γ , then δ(m ) = gG(m), where g ∈ {gm,τ S τ ∈ Im}, and

′ H(m ) = Hδ(m ) = HgG(m) = H ∩ GgG(m). ′ By Proposition 4.4.4, ′ −1 H(m ) = H ∩ GgG(m) = H ∩ gG(m)g , since G(ω) = 1 and we are identifying G(i) with Gi when i = 1, ⋯, n, we have

n −1 H = O O(H ∩ gi,τ Gigi,τ ) F, i=1 τ∈Ii

C ′ where F = π1 (Γ ). Then F is a free pro-C group by Proposition 3.4.3, whose rank is ′ ′ rF = SΓ S − ST S, where T is the maximal tree of Γ . To compute this number, put t = [G ∶ H] and ti = SHG~GiS. Then n ′ SΓ S = (n + 1)t + Q ti. i=1 ′ Since T is a ﬁnite tree ST S = 2SV (T )S − 1 = 2SV (Γ )S − 1; so

n ST S = 2(t + Q ti) − 1. i=1

Therefore, n rF = 1 − t + Q(t − ti). i=1

60 Bibliography

[1] RIBES, L., ‘Proﬁnite graphs and groups’, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics 66 (Springer, Berlin, 2017).

[2] RIBES, L. and ZALESSKII, P. A., ’Proﬁnite groups’, 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 40 (Springer, Berlin-Heidelberg, New York, 2010).

[3] SERRE, J.-P., ‘Trees’, 1st english edition, (Springer, Berlin, 1980).