Surface Subgroups of Graph Groups
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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. -
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. -
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. -
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. -
Kurosh Rank of Intersections of Subgroups of Free Products of Right-Orderable Groups Yago Antol´In, Armando Martino and Inga Schwabrow
Math. Res. Lett. c 2014 International Press Volume 21, Number 04, 649–661, 2014 Kurosh rank of intersections of subgroups of free products of right-orderable groups Yago Antol´ın, Armando Martino and Inga Schwabrow We prove that the reduced Kurosh rank of the intersection of two subgroups H and K of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of H and K. In particular, taking the fundamental group of a graph of groups with trivial vertex and edge groups, and its Bass–Serre tree, our theorem becomes the desired inequality of the usual strengthened Hanna Neumann conjecture for free groups. 1. Introduction Let H and K be subgroups of a free group F, and r(H)=max{0, rank(H) − 1}. It was an open problem dating back to the 1950s to find an optimal bound of the rank of H ∩ K in terms of the ranks of H and K. In [14], Hanna Neumann proved the following: (1) r(H ∩ K) ≤ 2 · r(H) · r(K). The Hanna Neumann conjecture says that (1) holds replacing the 2 with a 1. Later, in [15], Walter Neumann improved (1) to (2) r(Hg ∩ K) ≤ 2 · r(H) · r(K), g∈K\F/H where Hg = g−1Hg. The strengthened Hanna Neumann conjecture,intro- duced by Walter Neumann, says that (2) holds replacing the 2 with a 1. These two conjectures have received a lot of attention, and recently, Igor Mineyev [13] proved that both conjectures are true (independently and at the same time, Joel Friedman also proved these conjectures (see [5, 8])). -
Contractible N-Manifolds and the Double N-Space Property Pete Sparks University of Wisconsin-Milwaukee
University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations December 2014 Contractible n-Manifolds and the Double n-Space Property Pete Sparks University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Mathematics Commons Recommended Citation Sparks, Pete, "Contractible n-Manifolds and the Double n-Space Property" (2014). Theses and Dissertations. 643. https://dc.uwm.edu/etd/643 This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctorate of Philosophy in Mathematics at The University of Wisconsin-Milwaukee December, 2014 ABSTRACT CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks The University of Wisconsin-Milwaukee, 2014 Under the Supervision of Professor Craig Guilbault We are interested in contractible manifolds M n which decompose or split as n n n M = A ∪C B where A, B, C ≈ R or A, B, C ≈ B . We introduce a 4-manifold M containing a spine which can be written as A ∪C B with A, B, and C all collapsible 4 4 which in turn implies M splits as B ∪B4 B . From M we obtain a countably infinite 4 4 collection of distinct 4-manifolds all of which split as B ∪B4 B . -
Contractible N-Manifolds and the Double N-Space Property Pete Sparks University of Wisconsin-Milwaukee
University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations December 2014 Contractible n-Manifolds and the Double n-Space Property Pete Sparks University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Mathematics Commons Recommended Citation Sparks, Pete, "Contractible n-Manifolds and the Double n-Space Property" (2014). Theses and Dissertations. 643. https://dc.uwm.edu/etd/643 This Dissertation is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctorate of Philosophy in Mathematics at The University of Wisconsin-Milwaukee December, 2014 ABSTRACT CONTRACTIBLE n-MANIFOLDS AND THE DOUBLE n-SPACE PROPERTY by Pete Sparks The University of Wisconsin-Milwaukee, 2014 Under the Supervision of Professor Craig Guilbault We are interested in contractible manifolds M n which decompose or split as n n n M = A ∪C B where A, B, C ≈ R or A, B, C ≈ B . We introduce a 4-manifold M containing a spine which can be written as A ∪C B with A, B, and C all collapsible 4 4 which in turn implies M splits as B ∪B4 B . From M we obtain a countably infinite 4 4 collection of distinct 4-manifolds all of which split as B ∪B4 B . -
On the Intersection of Free Subgroups in Free Products of Groups
On the intersection of free subgroups in free products of groups Warren Dicks and S. V. Ivanov January 1, 2018 Dedicated to the memory of Prof. Charles Thomas Abstract Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let G = F ∗ ∗ Gi, the free product of F and all the Gi. i∈I Let F denote the set of all finitely generated subgroups H of G which g have the property that, for each g ∈ G and each i ∈ I, H ∩ Gi = {1}. By the Kurosh Subgroup Theorem, every element of F is a free group. For each free group H, the reduced rank of H, denoted ¯r(H), is defined as max{rank(H) − 1, 0} ∈ N ∪{∞} ⊆ [0, ∞]. To avoid the vacuous case, we make the additional assumption that F contains a non-cyclic group, and we define ¯r(H∩K) F σ := sup{¯r(H)· ¯r(K) : H, K ∈ and ¯r(H)· ¯r(K) 6= 0} ∈ [1, ∞]. We are interested in precise bounds for σ. In the special case where I is empty, Hanna Neumann proved that σ ∈ [1, 2], and conjectured that σ = 1; almost fifty years later, this interval has not been reduced. ∞ With the understanding that ∞−2 is 1, we define |L| θ := max{ |L|−2 : L is a subgroup of G and |L| 6= 2} ∈ [1, 3]. arXiv:math/0702363v1 [math.GR] 13 Feb 2007 Generalizing Hanna Neumann’s theorem, we prove that σ ∈ [θ, 2 θ], and, moreover, σ = 2 θ whenever G has 2-torsion. -
Locally Compact Group Topofogies on an Algebraic Free Product of Groups*
JOURNAL OF ALGEBRA 38, 393-397 (1976) Locally Compact Group Topofogies on an Algebraic Free Product of Groups* SIDNEY A. MORRIS AND PETER NICKOLA~ Department of Mathematics, University of New South Wales, Kensington, New South Wales, Australia Communicated by J. Tits Received March 11, 1974 It is proved that any homomorphism from a locally compact group G into a free products of groups with the discrete topalogy is either continuous, or trivial in the sense that the image of G lies wholly in a conjugate ‘of one of the factors of the free product. Hence, any locally compact group topology on a (nontrivial) free product of groups is discrete. 1. INTRODUCTION In Cl], Dudley proved the striking fact that any homomorphism from a locally compact group into a free group with the discrete topology is con- tinuous. Our aim is to generalize this result to the casewhere the range group is an arbitrary free product of groups. In fact, we show that any homomor- phism from a locally compact group G into a free product of groups with the discrete topology is either continuous, or trivial in the sensethat the image of G lies wholIy in a conjugate of one of the factors of the free product. Dudley proved his results by means of the introduction of “moms” into certain groups, free groups and free abelian groups in particular, and the exploitation of their special properties. His method, however, is not imme- diately applicable to the problem we consider, since a free product of groups is not generally normable. Our proof instead dependsheavily on the Kurosh subgroup theorem for free products and a theorem of Iwasawa [&I on the structure of connected locally compact groups. -
On Subgroups of Finite Complexity in Groups Acting on Trees
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 200 (2005) 1–23 www.elsevier.com/locate/jpaa On subgroups of finite complexity in groups acting on trees Mihalis Sykiotis∗,1 Department of Mathematics, University of Athens, Athens 15784, Greece Received 2 August2004; received in revised form 13 October2004 Available online 18 April 2005 Communicated by C.A. Weibel Abstract Let G be a group acting on a tree X.We show that some classical results concerning finitely generated subgroups of free groups, free products, and free-by-finite groups, remain valid if we replace finitely generated subgroups by tame subgroups or by subgroups of finite complexity. We also prove new results for tame subgroups and, more generally, for subgroups of finite complexity. © 2005 Published by Elsevier B.V. MSC: Primary: 20E06; 20E07; 20E08 1. Introduction Let G be a group acting on a tree X. Given a subgroup H of G, it is natural to ask if there is an invariantof H uniquely determined by the action of H on X, independent of the cardinality of any setof generatorsof H. Furthermore, the fact that in many cases (for example some vertex groups contain infinitely generated subgroups) the existence of infinitely generated subgroups of G is unavoidable, makes clear the need of finding such an invariant. One step in this direction has been done for subgroups of free products with the introduc- tion of the Kurosh rank (see [6,8] and [20] from which we have taken the term complexity).