View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UCL Discovery Geometric structures on negatively curved groups and their subgroups Samuel Brown A thesis submitted in partial fulfilment for the award of the degree of Doctor of Philosophy Department of Mathematics University College London Submitted August 2016 Revised January 2017 © Samuel Brown 2016 All rights reserved Declaration I, Samuel Brown, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. ................................................... 2 Abstract In this thesis, we investigate two explicit families of geometric structures that occur on hy- perbolic groups. After recalling some introductory material, we begin by giving an overview of the theory of special cube complexes, with a particular focus on properties of subgroups of hyperbolic special groups. We then describe an explicit algorithm, based on Stallings’ notion of folding for graphs, to construct a local isometry between cube complexes that represents the inclusion of a subgroup H G, and show that this terminates if and only if the subgroup ½ is quasiconvex. This provides a potential method by which quasiconvexity for various sub- groups could be verified. In the second part of the thesis, we investigate another family of geometric structures: negatively curved simplicial complexes. We show that groups satisfying a “uniform” C 0(1/6) small cancellation condition have such a structure, and then move on to prove a gluing theo- rem (with cyclic edge groups) for these complexes. Using this theorem, we extend the family of groups known to be CAT( 1) to include hyperbolic limit groups, hyperbolic graphs of free ¡ groups with cyclic edge groups, and more generally hyperbolic groups whose JSJ components are 2-dimensionally CAT( 1). ¡ Primary Supervisor: Dr Henry Wilton 3 Acknowledgements Studying for a PhD has been an immensely rewarding experience, but never an easy one. There are many people without whose support I could not have completed this thesis, and I am grateful for this opportunity to express my thanks. Most importantly, I would like to thank my supervisor, Dr Henry Wilton. It has been an enormous pleasure to spend the last four years working with Henry, and his mathematical guidance, moral support, and boundless optimism have kept me going throughout the last four years. I could not have hoped for a better experience of supervision. More broadly, I would like to express my gratitude to the wider geometric group theory community for welcoming me into their fold at many enjoyable conferences and seminars during my PhD. I am particularly grateful to those at the mathematics departments at Oxford and Cambridge, for their warm welcome to me during many visits over the years. My experience as a PhD student has been greatly enhanced by the work of both the aca- demic and the administrative staff in the UCL Maths Department, and so I would like to ex- press my gratitude to them. I would also like to take the opportunity to thank the EPSRC for funding my research, and all those who have inspired and nurtured my passion for mathe- matics throughout my education. On a less academic note, my life as a PhD student would have been far poorer without the friendship and entertainment offered by my colleagues in the KLB postgraduate office at UCL. They have helped keep me sane, and provided regular welcome distractions from my research. I wish them all the very best for the remainder of their PhD’s, and into the future. I would also like to thank all my other friends, both in London and further afield, for their support over the last four years. Lastly, I would like to thank my family for continuing to inspire me; and my partner Louise, whose unending love and friendship mean more to me than I can adequately express. 4 Contents Abstract 3 Acknowledgements 4 1 Introduction 8 2 Preliminaries 13 2.1 Graphs of groups and graphs of spaces . 13 2.1.1 Graphs . 13 2.1.2 Graphs of groups . 14 2.1.3 Graphs of spaces . 15 2.2 CAT(k) spaces . 17 2.2.1 The CAT(k) criterion . 18 2.2.2 Local isometries . 19 2.2.3 The Cartan-Hadamard Theorem . 19 2.2.4 Mk –complexes . 20 2.2.5 The link condition in simplicial complexes . 21 2.3 Cube complexes . 23 2.3.1 Local isometries of cube complexes . 25 2.4 Negative curvature and group theory . 26 2.4.1 Geometric actions and the Švarc–Milnor lemma . 26 2.4.2 Hyperbolic spaces and groups . 27 2.4.3 Relationships between the notions of negative curvature . 28 2.4.4 Torsion and the fundamental group . 30 2.4.5 Dimensions of groups . 31 2.5 Subgroups of negatively curved groups . 31 CONTENTS 6 2.5.1 Quasiconvexity . 31 2.5.2 Malnormality and finite width . 33 3 Special cube complexes 35 3.1 Cube complexes and their hyperplanes . 36 3.1.1 Sageev’s dual cube complex construction . 37 3.2 Right-angled Artin groups . 38 3.2.1 Salvetti complexes . 38 3.3 Special cube complexes . 40 3.4 Subgroup separability . 45 3.4.1 Topological motivation . 46 3.4.2 Free groups . 47 3.4.3 Canonical completion and retraction . 48 3.5 Quasiconvexity in cube complexes . 51 3.5.1 Convex subcomplexes of CAT(0) cube complexes . 53 3.5.2 Separability of quasiconvex subgroups . 54 3.6 Hyperplane subgroups . 55 3.7 VH complexes . 57 3.8 Further developments . 58 3.8.1 Hierarchies . 59 3.8.2 3-manifolds . 61 4 Folding cube complexes 64 4.1 Folding for graphs . 65 4.2 Graphs of graphs . 68 4.2.1 Lollipops . 69 4.3 Folding graphs of graphs . 70 4.3.1 Outline of the folding algorithm . 72 4.3.2 The folding moves . 73 4.3.3 The folding algorithm . 78 4.3.4 Termination, stabilisation and the limit complex . 81 4.3.5 Folding and presentations . 86 4.4 Folding for subgroups of finite width . 88 4.4.1 Motivation: quasiconvexity of finite width subgroups . 88 CONTENTS 7 4.4.2 Asymptotic injectivity radius . 92 4.4.3 Potential approach: asymptotic geometry of the vertex spaces . 93 4.4.4 Other potential approaches . 94 4.5 Higher dimensional folding . 95 5 Negatively curved metrics on small cancellation groups 97 5.1 Preliminaries . 98 5.1.1 Small cancellation conditions . 98 5.1.2 Geometry of regular polygons . 99 5.1.3 Singular polygons . 102 5.2 Proof of the main theorem . 102 6 A gluing theorem for negatively curved complexes 108 6.1 Preliminaries . 109 6.1.1 Metrizing graphs of spaces . 109 6.1.2 Thinness . 110 6.1.3 Excess angle and comparison simplices . 110 6.1.4 Hyperbolic triangles, quadrilaterals and annuli . 113 6.1.5 Transversality . 117 6.2 The gluing theorem . 120 6.3 Limit groups . 123 6.4 Further applications of the gluing theorem . 128 6.4.1 JSJ decompositions of torsion-free hyperbolic groups . 128 6.4.2 Graphs of free groups with cyclic edge groups . 131 6.5 Remarks . 132 Bibliography 134 Chapter 1 Introduction Indisputably one of the most outstanding achievements of twentieth or twenty-first century mathematics has been Bill Thurston’s geometrization program for three-dimensional mani- folds. First proposed by Thurston in the 1980s, the geometrization conjecture states that ev- ery 3-manifold can be decomposed by cutting along spheres and tori into pieces which admit one of precisely eight different geometries. Perhaps the richest and most interesting of these eight options is hyperbolic geometry, which describes manifolds whose universal cover is hy- perbolic 3-space H3. It turns out that any compact 3-manifold which is aspherical (contains no essential spheres) and atoroidal (contains no essential tori) is hyperbolic. This is known as Thurston’s hyperbolization theorem, proved by Thurston in the special case of Haken mani- folds (see [Thu86] and subsequent papers) and in general by Perelman as part of his resolution of the geometrization conjecture (see [Per02]). What the hyperbolization theorem hints at is some sort of underlying “genericity” of hyperbolic geometry; to ensure that a 3-manifold is hyperbolic, it suffices simply to rule out certain “obvious” ways in which it could fail to be so. The deep underlying fact is that this prevalence of negative curvature extends well beyond the realm of 3-dimensional geometry. Motivated by the geometry of hyperbolic manifolds, Mikhail Gromov defined in [Gro87] the notion of a word–hyperbolic (or simply hyperbolic) group. A hyperbolic group shares some characteristics with the fundamental group of a hyperbolic manifold; in particular, it pos- sesses an intrinsic notion of negative curvature. However the class of hyperbolic groups is much more general than the class of fundamental groups of hyperbolic 3-manifolds, and they are central objects of study in geometric group theory. 8 CHAPTER1.INTRODUCTION 9 As with hyperbolic manifolds, hyperbolic groups are known to have a certain genericity; Gromov himself observed that (roughly speaking) in group presentations with a fixed gener- ating set and q relations, the probability that the corresponding group is hyperbolic goes to 1 as the lengths of the relations go to infinity. However, a direct analogue of the hyperbolization theorem for hyperbolic groups remains elusive. One possible such analogue is proposed in the following question. Question 1.0.1. Let ¡ be a group which is type F and contains no Baumslag–Solitar sub- 1 groups. Is ¡ hyperbolic? It is worth unpicking this question to understand exactly how it relates to the hyper- bolization theorem.