DOCSLIB.ORG

Nielsen Realisation by Gluing: Limit Groups and Free Products

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nielsen Realisation by Gluing: Limit Groups and Free Products NIELSEN REALISATION BY GLUING: LIMIT GROUPS AND FREE PRODUCTS SEBASTIAN HENSEL AND DAWID KIELAK Abstract. We generalise the Karrass{Pietrowski{Solitar and the Nielsen realisation theorems from the setting of free groups to that of free prod- ucts. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel{Mosher and on the outer space of a free product of Guirardel{ Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained. 1. Introduction In its original form, the Nielsen realisation problem asks which finite sub- groups of the mapping class group of a surface can be realised as groups of homeomorphisms of the surface. A celebrated result of Kerckhoff [Ker1, Ker2] answers this in the positive for all finite subgroups, and even allows for realisations by isometries of a suitable hyperbolic metric. Subsequently, similar realisation results were found in other contexts, per- haps most notably for realising finite groups in Out(Fn) by isometries of a suitable graph (independently by [Cul], [Khr], [Zim]; compare [HOP] for a different approach). In this article, we begin to develop a relative approach to Nielsen realisa- tion problems. The philosophy here is that if a group G allows for a natural arXiv:1601.02187v3 [math.GR] 14 Feb 2018 decomposition into pieces, then Nielsen realisation for Out(G) may be re- duced to realisation in the pieces, and a gluing problem. In addition to just solving Nielsen realisation for finite subgroups of Out(G), such an approach yields more explicit realisations, which also exhibit the structure of pieces for G. We demonstrate this strategy for two classes of groups: free products and limit groups. In another article, we use the results presented here, together with the philosophy of relative Nielsen realisation, to prove Nielsen realisation for certain right-angled Artin groups ([HK]). The second-named author was supported by the SFB 701. 1 2 SEBASTIAN HENSEL AND DAWID KIELAK The early proofs of Nielsen realisation for free groups rely in a fundamental way on a result of Karrass{Pietrowski{Solitar [KPS], which states that every finitely generated virtually free group acts on a tree with finite edge and vertex stabilisers. In the language of Bass{Serre theory, it amounts to saying that such a virtually free group is a fundamental group of a finite graph of groups with finite edge and vertex groups. This result of Karrass{Pietrowski{Solitar in turn relies on the celebrated theorem of Stallings on groups with at least two ends [Sta1, Sta2]. Stallings' theorem states that any finitely generated group with at least two ends splits over a finite group, which means that it acts on a tree with a single edge orbit and finite edge stabilisers. Equivalently: it is a fundamental group of a graph of groups with a single edge and a finite edge group. In the first part of this article, we generalise these results to the setting of a free product A = A1 ∗ · · · ∗ An ∗ B in which we (usually) require the factors Ai to be finitely generated, and B to be a finitely generated free group. Consider any finite group H acting on A by outer automorphisms in a way preserving the given free-product decomposition, by which we mean that each element of H sends each sub- group Ai to some Aj (up to conjugation); note that we do not require the action of H to preserve B in any way. We then obtain a corresponding group extension 1 ! A ! A ! H ! 1 In this setting we prove (for formal statements, see the appropriate sections) Relative Stallings' theorem (Theorem 2.7): A splits over a finite group, in such a way that each Ai fixes a vertex in the associated action on a tree. Relative Karrass{Pietrowski{Solitar theorem (Theorem 4.1): A acts on a tree with finite edge stabilisers, and with each Ai fixing a vertex of the tree, and with, informally speaking, all other vertex groups finite. Relative Nielsen realisation theorem (Theorem 7.5): Suppose that we are given complete non-positively curved (i.e. locally CAT(0)) spaces Xi realising the induced actions of H on the factors Ai. Then the action of H can be realised by a complete non-positively curved space X; in fact X can be chosen to contain the Xi in an equivariant manner. We emphasise that such a relative Nielsen realisation is new even if all Ai are free groups, in which case it answers a question of Karen Vogtmann. The classical Nielsen realisation for graphs immediately implies that a finite subgroup H < Out(Fn) fixes points in the Culler{Vogtmann Outer Space (defined in [CV]), as well as in the complex of free splittings of Fn (which is a simplicial closure of Outer Space). As an application of the NIELSEN REALISATION BY GLUING: LIMIT GROUPS AND FREE PRODUCTS 3 work in this article, we similarly obtain fixed point statements (Corollar- ies 5.1 and 6.1) for the graph of relative free splittings defined by Handel and Mosher [HM], and the outer space of a free product defined by Guirardel and Levitt [GL]. In the last section of the paper we prove Theorem 8.12. Let A be a limit group, and let A ! A ! H be an extension of A by a finite group H. Then there exists a complete compact locally CAT(κ) space X realising the extension A, where κ = −1 when A is hyperbolic, and κ = 0 otherwise. When κ = −1, the space X is of dimension at most 2. This theorem is obtained by combining the classical Nielsen realisation theorems (for free, free-abelian and surface groups { see Theorems 8.2 to 8.4) with the existence of an invariant JSJ decomposition shown by Bumagin{ Kharlampovich{Myasnikov [BKM]. Note that, in general, having a graph of groups decomposition for a group G with CAT(0) vertex groups and virtually cyclic edge groups does not allow one to build a CAT(0) space for G to act on, and thus conclude that G is itself CAT(0); the JSJ decompositions of limit groups are however special in this respect, and the extra structure allows for the conclusion. This has been observed by Sam Brown in [Bro], where he developed techniques for building up a CAT(0) space for G to act on. Observe that we obtain optimal curvature bounds for our space X { it has been proved by Alibegovi´c{Bestvina[AB] that limit groups are CAT(0), and by Sam Brown [Bro] that a limit group is CAT(−1) if and only if it is hyperbolic. Also, taking H to be the trivial group gives a new (more direct) proof of the fact that limit groups are CAT(0). Throughout the paper, we are going to make liberal use of the standard terminology of graphs of groups. The reader may find all the necessary information in Serre's book [Ser]. We are also going to make use of standard facts about CAT(0) and non-positively curved (NPC) spaces, as well as more general CAT(κ) spaces; the standard reference here is the book by Bridson{ Haefliger [BH]. Acknowledgements. The authors would like to thank Karen Vogtmann for discussions and suggesting the statement of relative Nielsen realisation for free groups, Stefan Witzel for pointing out the work of Sam Brown, and the referee for extremely valuable comments. 2. Relative Stallings' theorem In this section we will prove the relative version of Stallings' theorem. Before we can begin with the proof, we need a number of definitions to 4 SEBASTIAN HENSEL AND DAWID KIELAK formalise the notion of a free splitting that is preserved by a finite group action. Convention. When talking about free factor decompositions A = A1 ∗ · · · ∗ An ∗ B of some group A, we will always assume that at least two of the factors fA1;:::;An;Bg are non-trivial. Definition 2.1. Suppose that φ: H ! Out(A) is a homomorphism with a finite domain. Let A = A1 ∗ · · · ∗ An ∗ B be a free factor decomposition of A. We say that this decomposition is preserved by H if and only if for every i and every h 2 H, there is some j such that h(Ai) is conjugate to Aj. We say that a factor Ai is minimal if and only if for any h 2 H the fact that h(Ai) is conjugate to Aj implies that j > i. Remark 2.2. Note that when the decomposition is preserved, we obtain an induced action H ! Sym(n) on the indices 1; : : : ; n. We may thus speak of the stabilisers StabH (i) inside H. Furthermore, we obtain an induced action StabH (i) ! Out(Ai) The minimality of factors is merely a way of choosing a representative of each H orbit in the action H ! Sym(n). Remark 2.3. Given an action φ: H ! Out(A), with φ injective and A with trivial centre, we can define A 6 Aut(A) to be the preimage of H = im φ under the natural map Aut(A) ! Out(A). We then note that A is an extension of A by H: 1 ! A ! A ! H ! 1 and the left action of H by outer automorphisms agrees with the left con- jugation action inside the extension A.
Load more

Recommended publications
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • University of Southampton Research Repository Eprints Soton
    University of Southampton Research Repository ePrints Soton Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk UNIVERSITY OF SOUTHAMPTON Deformation spaces and irreducible automorphisms of a free product by Dionysios Syrigos A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in the Faculty of Mathematics School of Social, Human, and Mathematical Sciences July, 2016 Sthn oikogèneia mou gia thn υποστήρixh pou mou èqei d¸sei UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF MATHEMATICS SCHOOL OF SOCIAL, HUMAN, AND MATHEMATICAL SCIENCES Doctor of Philosophy by Dionysios Syrigos The (outer) automorphism group of a finitely generated free group Fn, which we denote by Out(Fn), is a central object in the fields of geometric and combinatorial group theory. My thesis focuses on the study of the automorphism group of a free product of groups. As every finitely generated group can be written as a free product of finitely many freely indecomposable groups and a finitely generated free group (Grushko’s Theorem) it seems interesting to study the outer automorphism group of groups that split as a free product of simpler groups.
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • HAMILTONICITY in CAYLEY GRAPHS and DIGRAPHS of FINITE ABELIAN GROUPS. Contents 1. Introduction. 1 2. Graph Theory: Introductory
    HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. MARY STELOW Abstract. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. Several results are then pre- sented: firstly, it is shown that if G is an abelian group, then G has a Cayley digraph with a Hamiltonian cycle. It is then proven that every Cayley di- graph of a Dedekind group has a Hamiltonian path. Finally, we show that all Cayley graphs of Abelian groups have Hamiltonian cycles. Further results, applications, and significance of the study of Hamiltonicity of Cayley graphs and digraphs are then discussed. Contents 1. Introduction. 1 2. Graph Theory: Introductory Definitions. 2 3. Cayley Graphs and Digraphs. 2 4. Hamiltonian Cycles in Cayley Digraphs of Finite Abelian Groups 5 5. Hamiltonian Paths in Cayley Digraphs of Dedekind Groups. 7 6. Cayley Graphs of Finite Abelian Groups are Guaranteed a Hamiltonian Cycle. 8 7. Conclusion; Further Applications and Research. 9 8. Acknowledgements. 9 References 10 1. Introduction. The topic of Cayley digraphs and graphs exhibits an interesting and important intersection between the world of groups, group theory, and abstract algebra and the world of graph theory and combinatorics. In this paper, I aim to highlight this intersection and to introduce an area in the field for which many basic problems re- main open.The theorems presented are taken from various discrete math journals. Here, these theorems are analyzed and given lengthier treatment in order to be more accessible to those without much background in group or graph theory.
    [Show full text]
  • A Rank Formula for Acylindrical Splittings
    A RANK FORMULA FOR ACYLINDRICAL SPLITTINGS RICHARD WEIDMANN Dedicated to Michel Boileau on the occasion of his 60th birthday Abstract. We prove a rank formula for arbitrary acylindrical graphs of groups and deduced that the the Heegaard genus of a closed graph manifold can be bounded by a linear function in the rank of its fundamental group. Introduction Grushko's Theorem states that the rank of groups is additive under free products, i.e. that rank A ∗ B = rank A + rank B. The non-trivial claim is that rank A ∗ B ≥ rank A ∗ rank B. In the case of amalgamated products G = A ∗C B with finite amalgam a similar lower bound for rank G can be given in terms of rank A, rank B and the order of C [We4]. For arbitrary splittings this is no longer true. It has first been observed by G. Rosenberger [Ro] that the naive rank for- mula rank G ≥ rank A + rank B − rank C does not hold for arbitrary amalga- mated products G = A ∗C B, in fact Rosenberger cites a class of Fuchsian groups as counterexamples. In [KZ] examples of Coxeter-groups where exhibited where rank G1 ∗Z2 G2 = rank G1 = rank G2 = n with arbitrary n. In [We2] examples of groups Gn = An ∗C Bn are constructed such that rank An ≥ n, rank Bn ≥ n and rank C = rank Gn = 2. These examples clearly show that no non-trivial analogue of Grushko's Theorem holds for arbitrary splittings. If the splitting is k-acylindrical however, this situation changes. This has been shown in [We3] for 1-acylindrical amalgamated products where it was also claimed that a similar result hold for arbitrary splittings.
    [Show full text]
  • The Monadic Second-Order Logic of Graphs Xvi: Canonical Graph Decompositions
    Logical Methods in Computer Science Vol. 2 (2:2) 2006, pp. 1–46 Submitted Jun. 24, 2005 www.lmcs-online.org Published Mar. 23, 2006 THE MONADIC SECOND-ORDER LOGIC OF GRAPHS XVI: CANONICAL GRAPH DECOMPOSITIONS BRUNO COURCELLE LaBRI, Bordeaux 1 University, 33405 Talence, France e-mail address: [email protected] Abstract. This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decom- position and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs having the same cycle matroid as a given 2-connected graph can be defined from this graph by Monadic Second-Order formulas. 1. Introduction Hierarchical graph decompositions are useful for the construction of efficient algorithms, and also because they give structural descriptions of the considered graphs. Cunningham and Edmonds have proposed in [18] a general framework for defining decompositions of graphs, hypergraphs and matroids. This framework covers many types of decompositions. Of particular interest is the split decomposition of directed and undirected graphs defined by Cunningham in [17]. A hierarchical decomposition of a certain type is canonical if, up to technical details like vertex labellings, there is a unique decomposition of a given graph (or hypergraph, or matroid) of this type. To take well-known examples concerning graphs, the modular decom- position is canonical, whereas, except in particular cases, there is no useful canonical notion of tree-decomposition of minimal tree-width.
    [Show full text]
  • Foldings, Graphs of Groups and the Membership Problem
    FOLDINGS, GRAPHS OF GROUPS AND THE MEMBERSHIP PROBLEM ILYA KAPOVICH, RICHARD WEIDMANN, AND ALEXEI MYASNIKOV Abstract. We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Ap- plications include coherent right-angled Artin groups and coherent solvable groups. 1. Introduction The idea of using foldings to study group actions on trees was introduced by Stallings in a seminal paper [45], where he applied foldings to investigate free groups. Free groups are exactly those groups that admit free actions on simplicial trees. Later Stallings [46] offered a way to extend these ideas to non-free actions of groups on graphs and trees. Bestvina and Feighn [5] gave a systematic treatment of Stallings’ approach in the con- text of graphs of groups and applied this theory to prove a far-reaching generalization of Dunwoody’s accessibility result. Later Dunwoody [20] refined the theory by introducing vertex morphism. Dunwoody [21] used foldings to construct a small unstable action on an R-tree. Some other applications of foldings in the graph of groups context can be found in [39, 42, 43, 17, 19, 27, 28, 13, 12, 29]. In this paper we develop a combinatorial treatment of foldings geared towards more com- putational questions. In particular we are interested in the subgroup membership problem and in computing algorithmically the induced splitting for a subgroup of the fundamental group of a graph of groups. Recall that a finitely generated group G = hx1, .
    [Show full text]
  • A Rank Formula for Amalgamated Products with Finite Amalgam
    A rank formula for amalgamated products with finite amalgam Richard Weidmann 1 Introduction Grushko’s theorem [4] says that the rank of a group is additive under free products, i.e. that rank A ∗ B = rank A + rank B. For amalgamated products no such formula exists unless we make strong assumptions on the amalgamation [11]; in fact for any n there exists an amalga- mated product Gn = An ∗F2 Bn such that rank Gn = 2 and rank An, rank Bn ≥ n [10]. Of course F2, the free group of rank 2, is a very large group. Thus it makes sense to ask whether the situation is better if the amalgam is a small group. In this note we give a positive answer to this question in the case that the amalgam is finite. For a finite group C denote by l(C) the length of the longest strictly as- cending chain of non-trivial subgroups. Thus l(C) = 1 iff C is a cyclic group of prime order. We show the following: Theorem Suppose that G = A ∗C B with C finite. Then 1 rank G ≥ (rank A + rank B). 2l(C)+1 The theorem also holds for amalgamated products with infinite amalgam provided that there is an upper bound for the length of any strictly ascending chain of subgroups of C. Such groups must be torsion groups; examples are the Tarski monsters exhibited by A. Yu Ol’shanskii [7]. The proof is related to the proof of Dunwoody’s version [3] of Linnell’s [6] accessibility result for finitely generated groups.
    [Show full text]
  • Applications of Foldings of A-Graphs
    Applications of foldings of A-graphs Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Christian-Albrechts-Universität zu Kiel vorgelegt von Ederson Ricardo Fruhling Dutra Kiel Februar 2018 1. Gutachter: Prof. Dr. Richard Weidmann 2. Gutachter: Prof. Dr. Hartmut Weiß Tag der mündlichen Prüfung: 05.02.2018 Zum Druck genehmigt: 05.02.2018 gez. Prof. Dr. Natascha Oppelt, Dekanin Abstract This thesis gives some applications of the theory of folding of A-graphs. In the first appli- cation, we investigate the relationship between the bridge number of a knot k, a geometrically defined invariant of the knot, and the meridional rank of the knot group G(k). We give a new proof that these two invariants coincide if k is a torus knot. With similar techniques we show that the same holds if k is a knot whose complement is a graph manifold. Next, we study killers of knot groups. We prove that the group G(k) of any cable knot k contains infinitely many killers, none of which is the automorphic image of another. As a corollary we show that if k is a knot whose complement is a graph manifold, then G(k) has the same property. Finally, we study generating sets of planar discontinuous groups. We show for a large class of planar discontinuous groups that there are only finitely many Nielsen equivalence classes of generating tuples. iv Zusammenfassung Diese Thesis beinhaltet einige Anwendungen der Faltungstheorie von Graphen von Gruppen. Dabei beschäftigen wir uns in der ersten Anwendung mit der Beziehung der sogenannten Brückenzahl, einer geometrischen Invarianten eines Knoten k, und dem sogenannten merdialem Rang der assoziierten Knotengruppe G(k).
    [Show full text]
  • General Topology. Part 1: First Concepts
    A Review of General Topology. Part 1: First Concepts Wayne Aitken∗ March 2021 version This document gives the first definitions and results of general topology. It is the first document of a series which reviews the basics of general topology. Other topics such as Hausdorff spaces, metric spaces, connectedness, and compactness will be covered in further documents. General topology can be viewed as the mathematical study of continuity, con- nectedness, and related phenomena in a general setting. Such ideas first arise in particular settings such as the real line, Euclidean spaces (including infinite di- mensional spaces), and function spaces. An initial goal of general topology is to find an abstract setting that allows us to formulate results concerning continuity and related concepts (convergence, compactness, connectedness, and so forth) that arise in these more concrete settings. The basic definitions of general topology are due to Fr`echet and Hausdorff in the early 1900's. General topology is sometimes called point-set topology since it builds directly on set theory. This is in contrast to algebraic topology, which brings in ideas from abstract algebra to define algebraic invariants of spaces, and differential topology which considers topological spaces with additional structure allowing for the notion of differentiability (basic general topology only generalizes the notion of continuous functions, not the notion of dif- ferentiable function). We approach general topology here as a common foundation for many subjects in mathematics: more advanced topology, of course, including advanced point-set topology, differential topology, and algebraic topology; but also any part of analysis, differential geometry, Lie theory, and even algebraic and arith- metic geometry.
    [Show full text]
  • Notes on 3-Manifolds
    CHAPTER 1: COMBINATORIAL FOUNDATIONS DANNY CALEGARI Abstract. These are notes on 3-manifolds, with an emphasis on the combinatorial theory of immersed and embedded surfaces, which are being transformed to Chapter 1 of a book on 3-Manifolds. These notes follow a course given at the University of Chicago in Winter 2014. Contents 1. Dehn’s Lemma and the Loop Theorem1 2. Stallings’ theorem on ends, and the Sphere Theorem7 3. Prime and free decompositions, and the Scott Core Theorem 11 4. Haken manifolds 28 5. The Torus Theorem and the JSJ decomposition 39 6. Acknowledgments 45 References 45 1. Dehn’s Lemma and the Loop Theorem The purpose of this section is to give proofs of the following theorem and its corollary: Theorem 1.1 (Loop Theorem). Let M be a 3-manifold, and B a compact surface contained in @M. Suppose that N is a normal subgroup of π1(B), and suppose that N does not contain 2 1 the kernel of π1(B) ! π1(M). Then there is an embedding f :(D ;S ) ! (M; B) such 1 that f : S ! B represents a conjugacy class in π1(B) which is not in N. Corollary 1.2 (Dehn’s Lemma). Let M be a 3-manifold, and let f :(D; S1) ! (M; @M) be an embedding on some collar neighborhood A of @D. Then there is an embedding f 0 : D ! M such that f 0 and f agree on A. Proof. Take B to be a collar neighborhood in @M of f(@D), and take N to be the trivial 0 1 subgroup of π1(B).
    [Show full text]
  • Applications of Order Treesin Infinite Graphs
    Max Pitz: Applications of order trees in infinite graphs 27. August 2020, ALGOS2020 Slides available under https://www.math.uni-hamburg.de/home/pitz/ §0: T -graphs: Definition and examples §1: T -graphs, colouring number and forbidden minors §2: T -graphs and wqo of infinite graphs §3: Halin’s end degree conjecture §0: T -graphs: Definition and examples Let’s agree on the following notation regarding order trees: • Order tree: A partially ordered set (T, ≤) with unique minimal element (called the root) and all sub- 0 0 sets of the form dte = dteT := {t ∈ T : t ≤ t} are well-ordered. Write btc := {t0 ∈ T : t ≤ t0}. • Branch: A maximal chain in T (well-ordered). • Height: The height of T is the supremum of the order types of its branches. The height of a point t ∈ T is the order type of d˚te := dte \ {t}. • Level: The set T i of all points at height i is the ith level of T , and write T <i := S T j : j < i . • Successors and limits: If t < t0, we write [t, t0] = {x: t ≤ x ≤ t0} etc. If t < t0 but there is no point between t and t0, we call t0 a successor of t and t the predecessor of t0; if t is not a successor of any point it is called a limit. §0: T -graphs: Definition and examples Let’s agree on the following notation regarding order trees: • Order tree: A partially ordered set (T, ≤) with unique minimal element (called the root) and all subsets of the form 0 0 0 0 dte = dteT := {t ∈ T : t ≤ t} are well-ordered.
    [Show full text]