Effectiveness in Stallings' Proof of Grushko's Theorem
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III March 5, 2002 Abstract These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller. Contents 1 Free groups and presentations 3 1.1 Free groups . 3 1.2 Presentations by generators and relations . 7 1.3 Dehn’s fundamental problems . 9 1.4 Homomorphisms . 10 1.5 Presentations and fundamental groups . 12 1.6 Tietze transformations . 14 1.7 Extraction principles . 15 2 Construction of new groups 17 2.1 Direct products . 17 2.2 Free products . 19 2.3 Free products with amalgamation . 21 2.4 HNN extensions . 24 3 Properties, embeddings and examples 27 3.1 Countable groups embed in 2-generator groups . 27 3.2 Non-finite presentability of subgroups . 29 3.3 Hopfian and residually finite groups . 31 4 Subgroup Theory 35 4.1 Subgroups of Free Groups . 35 4.1.1 The general case . 35 4.1.2 Finitely generated subgroups of free groups . 35 4.2 Subgroups of presented groups . 41 4.3 Subgroups of free products . 43 4.4 Groups acting on trees . 44 5 Decision Problems 45 5.1 The word and conjugacy problems . 45 5.2 Higman’s embedding theorem . 51 1 5.3 The isomorphism problem and recognizing properties . 52 2 Chapter 1 Free groups and presentations In introductory courses on abstract algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself. -
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. -
Classification and Statistics of Finite Index Subgroups in Free Products
CLASSIFICATION AND STATISTICS OF FINITE INDEX SUBGROUPS IN FREE PRODUCTS Thomas W. Muller¨ and Jan-Christoph Schlage-Puchta 1. introduction ∗e For positive integers e and m denote by Cm the free product of e copies of the cyclic group of order m, and let Fr be the free group of rank r. Given integers r, t ≥ 0, distinct primes p1, . , pt, and positive integers e1, . , et, let ∗e1 ∗et Γ = Cp1 ∗ · · · ∗ Cpt ∗ Fr. (1) By the Kurosh subgroup theorem, a finite index subgroup ∆ ≤ Γ is again of the same ∼ ∗λ1 ∗λt form, that is, ∆ = Cp1 ∗ · · · ∗ Cpt ∗ Fµ with non-negative integers λ1, . , λt, µ. An Euler characteristic computation shows that the latter parameters are related to the index (Γ : ∆) via the relation X 1 X 1 λ 1 − + µ − 1 = (Γ : ∆) 1 − + r − 1 . (2) j p p j j j j The tuple τ(∆) := (λ1, . , λt; µ) is called the (isomorphism) type of ∆. The principal theme of the present paper is the enumeration of finite index subgroups ∆ in Γ under restrictions on τ(∆). In particular, we shall discuss, for Γ as above, the following three basic problems. (I) (Realization) Which abstract groups admitted by the Kurosh subgroup theorem are realized as finite index subgroups of Γ? (II) (Asymptotics) Find natural deformation conditions on τ ∈ Rt+1 implying an interesting asymptotic behaviour of the function sτ (Γ) counting the number of finite index subgroups in Γ of type τ. (III) (Distribution) What can we say about the distribution of isomorphism types for subgroups of index n in Γ (with respect to various weight distributions) as n tends to infinity? The motivation for these questions comes from three main sources: number theory, geometric function theory, and the theory of subgroup growth. -
3-Manifolds with Subgroups Z Z Z in Their Fundamental Groups
Pacific Journal of Mathematics 3-MANIFOLDS WITH SUBGROUPS Z ⊕ Z ⊕ Z IN THEIR FUNDAMENTAL GROUPS ERHARD LUFT AND DENIS KARMEN SJERVE Vol. 114, No. 1 May 1984 PACIFIC JOURNAL OF MATHEMATICS Vol 114, No. 1, 1984 3-MANIFOLDS WITH SUBGROUPS ZΦZΦZ IN THEIR FUNDAMENTAL GROUPS E. LUFT AND D. SJERVE In this paper we characterize those 3-manifolds M3 satisfying ZΘZΘZC ^i(Λf). All such manifolds M arise in one of the following ways: (I) M = Mo # R, (II) M= Mo # R*, (III) M = Mo Uθ R*. Here 2 Λf0 is any 3-manifold in (I), (II) and any 3-manifold having P compo- nents in its boundary in (III). R is a flat space form and R* is obtained from R and some involution t: R -> R with fixed points, but only finitely many, as follows: if C,,..., Cn are disjoint 3-cells around the fixed points then R* is the 3-manifold obtained from (R - int(C, U UQ))/ί by identifying some pairs of projective planes in the boundary. 1. Introduction. In [1] it was shown that the only possible finitely generated abelian subgroups of the fundamental groups of 3-manifolds are Zn9 Z θ Z2, Z, Z θ Z and Z θ Z θ Z. The purpose of this paper is to 3 characterize all M satisfying ZΘZΘZC πx(M). To explain this characterization recall that the Bieberbach theorem (see Chapter 3 of [8]) implies that if M is a closed 3-dimensional flat space form then ZΘZΘZC πx(M). We let M,,... 9M6 denote the 6 compact connected orientable flat space forms in the order given on p. -
Surface Subgroups of Graph Groups
Surface Subgroups of Graph Groups Herman Servatius Carl Droms College of the Holy Cross James Madison University Worcester, MA 01610 Harrisonburg, Va. 22807 Brigitte Servatius Worcester Polytechnic Institute Worcester, Ma. 01609 Abstract Let Γ = (V; E) be a graph with vertex set V and edge set E. The graph group based on Γ, FΓ, is the group generated by V , with defining relations xy = yx, one for each pair (x; y) of adjacent vertices in Γ. For n 3, the n-gon is the graph with n vertices, v ; : : : ; v , and n ≥ 1 n edges (vi; vi+1), indices modulo n. In this article we will show that if Γ has a full subgraph which is isomorphic to an n-gon, then the commutator subgroup of FΓ, FΓ0 , has a subgroup which is isomorphic to the fundamental group of the orientable surface of genus 1 + (n n 3 − 4)2 − . So, in particular, the graph group of the pentagon contains a sub- group which is isomorphic to the group of the five-holed torus. As an application, we note that this implies that many artin groups contain surface groups, see [4]. We also use this result to study the com- mutator subgroups of certain graph groups, continuing the study of subgroups of graph groups begun in [2] and [6]. We show that FΓ0 is free if and only if Γ contains no full subgraph isomorphic to any n-gon with n 4, which is an improvement on a theorem in [1]. We also ≥ show that if Γ contains no full squares, then FΓ0 is a graph group if and only if it is free; this shows that there exist graphs groups whose commutator subgroups are not graph groups. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups . -
Surface Subgroups of Graph Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 3, July 1989 SURFACE SUBGROUPS OF GRAPH GROUPS HERMAN SERVATIUS, CARL DROMS, AND BRIGITTE SERVATIUS (Communicated by Warren J. Wong) Abstract. Given a graph T , define the group Fr to be that generated by the vertices of T, with a defining relation xy —yx for each pair x, y of adjacent vertices of T. In this article, we examine the groups Fr- , where the graph T is an H-gon, (n > 4). We use a covering space argument to prove that in this case, the commutator subgroup Ff contains the fundamental group of the orientable surface of genus 1 + (n - 4)2"-3 . We then use this result to classify all finite graphs T for which Fp is a free group. To each graph T = ( V, E), with vertex set V and edge set E, we associate a presentation PT whose generators are the elements of V, and whose rela- tions are {xy = yx\x ,y adjacent vertices of T). PT can be regarded as the presentation of a &-algebra kT, of a monoid Mr, or of a group Fr, called a graph group. These objects have been previously studied by various authors [2-8]. Graph groups constitute a subclass of the Artin groups. Recall that an Artin group is defined by a presentation whose relations all take the form xyx■ ■■= yxy ■• • , where the two sides have the same length n > 1, and there is at most one such relation for any pair of generators. To each such presentation we can associate a labeled graph T, which has a vertex for each generator, and for each relation xyx ■■ ■ = yxy • • • , an edge joining x and y and labeled " n ", where n is the length of each side of the relation. -
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. -
Exercise Set 6
Exercise set 6 Groups acting on trees Due by May 21 You can upload your solution to one exercise to sam-up.math.ethz.ch by May 21st. You are encouraged to work in pairs, which allows you to work together on two problems and get both solutions corrected. Solutions will be presented during the exercise class of May 27th. Bass{Serre trees If G is the fundamental group of a graph of groups, then the tree constructed in Theorem 8.10 is called the Bass{Serre tree of G. With these exercises, we get familiar with the construction. Exercise 1. Let G be a free product with amalgamation (respectively, an HNN extension). Express G as the fundamental group of a graph of groups, and show that its Bass{Serre tree coincides with the tree obtained via Theorem 6.12 (respectively, Theorem 7.14). Exercise 2. Consider the two GBS groups from Exercise 7 in Exercise set 5. Describe their Bass{Serre trees: define them, draw pictures, and compute the degrees of their vertices. Finite vertex groups The goal of this section is to prove the following application of the Fundamental Theorem of Bass{Serre theory, which vastly generalizes Exercise 2 in Exercise set 3. Proposition (Bass{Serre). Let G be the fundamental group of a finite connected graph of groups with finite vertex groups. Then G is virtually free. Remark. Time permitting, by the end of the lecture we will also prove the converse. The following exercise, which is where Bass{Serre theory comes into play, tells you where to find the free subgroup. -
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, . -
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. -
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).