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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. -
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). -
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). -
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. -
Unrecognizability of Manifolds
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 141 (2006) 325–335 www.elsevier.com/locate/apal Unrecognizability of manifolds A.V. Chernavskya,∗,V.P.Leksineb a IITP RAS, Russian Federation b State Pedagogical Institute of Kolomna, Russian Federation Available online 18 January 2006 Abstract We presentamodernized proof, with a modification by M.A. Shtan’ko, of the Markov theorem on the unsolvability of the homeomorphy problem for manifolds. We then discuss a proof of the S.P. Novikov theorem on the unrecognizability of spheres Sn for n ≥ 5, from which we obtain a corollary about unrecognizability of all manifolds of dimension at least five. An analogous argument then proves the unrecognizability of stabilizations (i.e. the connected sum with 14 copies of S2 × S2)ofallfour- dimensional manifolds. We also give a brief overview of known results concerning algorithmic recognizability of three-dimensional manifolds. c 2005 Elsevier B.V. All rights reserved. Keywords: Unsolvability of the recognition problem for PL manifolds; Adian series of group presentations; Markov manifolds 1. Introduction A.A. Markov’s influential paper “The unsolvability of the homeomorphy problem” [17]wasaconsiderable accomplishment, as it was the first time that the techniquesdeveloped in algebra for undecidability problems had been carried over to topology. The paper’s main result is the following: Theorem (Markov [17]). For every natural number n > 3 one can create an n-manifold Mn such that the problem of homeomorphy of manifolds to Mn is undecidable. This theorem has as corollaries the undecidability of the following three problems: the problem of homeomorphy of n-manifolds for n > 3; the problem of homeomorphy for polyhedra of degree no higher than n for n > 3; the general problem of homeomorphy. -
Combinatorial Group Theory (Pdf)
Combinatorial Group Theory George D. Torres Spring 2021 1 Introduction 2 2 Free Products 3 2.1 Free Groups . .3 2.2 Residual Finiteness . .6 2.3 Free Products . .8 2.4 Free Products with Amalgamation and HNN Extensions . 14 These are lecture notes from the Spring 2021 course M392C Combinatorial Group Theory at the University of Texas at Austin taught by Prof. Cameron Gordon. The recommended prerequisites for following these notes are a first course in algebraic topology and an undergraduate level knowledge of group theory. Not strictly re- quired, but beneficial, is a familiarity with low-dimensional topology. Note: the theorem/definition/lemma etc. numbers in these notes don’t necessarily coincide with those in Cameron’s notes. Please email me at [email protected] with any typos and suggestions. Any portions marked TODO are on my to-do list to finish time permitting. Last updated March 4, 2021 1 1. Introduction v (from the syllabus) We will discuss some topics in combinatorial group theory, withan emphasis on topological methods and applications. The topics covered will include:free groups, free products, free products with amalgamation, HNN extensions, Grushko’sTheorem, group presentations, Tietze transformations, Dehn’s decision problems: the word, conjugacy and isomorphism problems, residual finiteness, coherence, 3-manifoldgroups, 1-relator groups, the Andrews-Curtis Conjecture, recursive presentations, the Higman Embedding Theorem, the Adjan- Rabin Theorem, the unsolvability of the homeomorphism problem for manifolds of dimension ≥ 4. The association X π1(X) defines a useful relationship between topology and group theory. For example, covering spaces of a space X correspond to subgroups of π1(X). -
James K. Whittemore· Lectures in Mathematics Given at Yale University GROUP THEORY and THREE-DIMENSIONAL MANIFOLDS
James K. Whittemore· Lectures in Mathematics given at Yale University GROUP THEORY AND THREE-DIMENSIONAL MANIFOLDS by John Stallings New Haven and London, Yale University Press, 1971 1 BACKGROUND AND SIGNIFICANCE OF THREE-DIMENSIONAL MANIFOLDS I.A Introduction The study of three-dimensional manifolds has often interacted with a certain stream of group theory, which is concerned with free groups, free products, finite presentations of groups, and simUar combinatorial mat ters. Thus, Kneser's fundamental paper [4] had latent implications toward Grushko's Theorem [8]; one of the sections of that paper dealt with the theorem that, if a manifold's fundamental group is a fre.e product, then the manifold exhibits this geometrically, being divided into two regions by a sphere with appropriate properties. Kneser's proof is fraught with geo metric hazards, but one of the steps, consisting of modifying the I-skele ton of the manifold and dividing it up, contains obscurely something like Grushko's ,Theorem: that a set of ~enerators of a free product can be modified in a certain st"mple way so as to be the union of sets of genera tors of the factors. Similarly, in the sequence of theorems by PapakyriakopoUlos, the Loop ~eorem [14], Def1:n's Lemma, and the ~here ~eorem [15], there are implicit facts about group theory, which form the \~tn. subject of these If.'' chapt~rs. " ) Philosophically speaking, the depth and beauty of 3~manifold theory is, it seems to me, mainly due to the fact that its theorems have offshoots that eventually blossom in a different subject, namely group theory. -
Grushko's Theorem on Free Products, a Topological Proof
Grushko's Theorem on free products, a Topological Proof A minor thesis submitted to Indian Institute of Science Education and Research, Pune in partial fulfillment of the requirements for the Mathematics PhD Degree Program Minor Thesis Supervisor: Dr. Tejas Kalelkar by Makarand Sarnobat March, 2014 Indian Institute of Science Education and Research, Pune Dr. Homi Bhabha Road, Pashan, Pune, India 411008 This is to certify that this minor thesis entitled \Grushko's Theorem on free products, a Topological Proof" submitted towards the partial fulfillment of the Mathematics PhD degree program at the Indian Institute of Science Education and Research Pune, represents work carried out by Makarand Sarnobat under the supervision of Dr. Tejas Kalelkar. Dr. Tejas Kalelkar Dr. A. Raghuram Minor Thesis Supervisor PhD Thesis Supervisor Dr. B.S.M. Rao Dean PhD Program 1 Introduction This is a paper written by John R. Stallings titled \A topological proof of Grusko's Theorem on free products" which was published in Math. Zeitschr in the year 1965. It uses no more than basic topology and combina- torics to give a proof of an important theorem in Group Theory. One of the main advantages of this paper, other than the fact that it is a simple proof, is that the proof is algorithmic in nature. The main result proved in this paper is the following: Grushko's Theorem: Let φ :Γ ! ∗αΠα be a homomorphism of the free group Γ onto the free product of groups fΠαg: Then Γ is itself a free product, Γ = ∗αΓα, such that φ(Γα) ⊆ Πα. Grushko's Theorem has some interesting implications. -
On Accessibility for Pro-P Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Apollo On accessibility for pro-p groups Gareth Wilkes Centre for Mathematical Studies, University of Cambridge Wilberforce Rd, Cambridge CB3 0WA Abstract We define the notion of accessibility for a pro-p group. We prove that finitely generated pro-p groups are accessible given a bound on the size of their finite subgroups. We then construct a finitely generated inaccessible pro-p group, and also a finitely generated inaccessible discrete group which is residually p-finite. Keywords: Pro-p groups, accessibility, graphs of groups 1. Introduction A common thread in mathematics is the search for conditions which tame wild objects in some way and allow deeper analysis of their structure. In the case of groups finiteness properties, finite dimension properties, and accessibility all fall into this thread. Accessibility is the name given to the following concept. Given a group, it may be possible to split the group over a finite subgroup into pieces which could be expected to be somehow `simpler'. For a notion of `simpler' to be of much help, this process should terminate|it should not be possible to continue splitting the group ad infinitum. Once we have split as far as possible, the remaining pieces are one-ended and amenable to further study|in this way accessibility is the foundation of JSJ theory for groups. We adopt the following definition of accessibility, which is well known to be equivalent to the original classical definition of Wall [Wal71, SW79] in the case of (finitely generated) discrete groups. -
3-MANIFOLD GROUPS Introduction in This Paper We Give an Overview Of
3-MANIFOLD GROUPS MATTHIAS ASCHENBRENNER, STEFAN FRIEDL, AND HENRY WILTON Abstract. We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise. Introduction In this paper we give an overview of properties of fundamental groups of compact 3-manifolds. This class of groups sits between the class of fundamental groups of surfaces, which for the most part are well understood, and the class of fundamen- tal groups of higher dimensional manifolds, which are very badly understood for the simple reason that given any finitely presented group π and any n ≥ 4, there exists a closed n-manifold with fundamental group π. (See [CZi93, Theorem 5.1.1] or [ST80, Section 52] for a proof.) This basic fact about high-dimensional mani- folds is the root of many problems; for example, the unsolvability of the isomor- phism problem for finitely presented groups [Ad55, Rab58] implies that closed manifolds of dimensions greater than three cannot be classified [Mav58, Mav60]. The study of 3-manifold groups is also of great interest since for the most part, 3-manifolds are determined by their fundamental groups. More precisely, a closed, irreducible, non-spherical 3-manifold is uniquely determined by its fundamental group (see Theorem 2.3). Our account of 3-manifold groups is based on the following building blocks: (1) If N is an irreducible 3-manifold with infinite fundamental group, then the Sphere Theorem (see (C.1) below), proved by Papakyriakopoulos [Pap57a], implies that N is in fact an Eilenberg{Mac Lane space.