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Detecting Topological Properties of Boundaries of Hyperbolic Groups

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Detecting Topological Properties of Boundaries of Hyperbolic Groups Detecting topological properties of boundaries of hyperbolic groups Benjamin Barrett Department of Pure Mathematics and Mathematical Statistics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Trinity College November 2018 Detecting topological properties of boundaries of hyperbolic groups Benjamin Barrett In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as Gromov hyperbolicity. Such groups are surprisingly common: there is a sense in which a random group admits such an action, as do some groups of classical interest, such as fundamental groups of closed Riemannian manifolds with negative sectional curvature. If a group admits an action on a Gromov hyperbolic space then large scale properties of the space give useful invariants of the group. One particularly natural large scale property used in this way is the Gromov boundary. The Gromov boundary of a hyperbolic group is a compact metric space that is, in a sense, approximated by spheres of large radius in the Cayley graph of the group. The technical results contained in this thesis are effective versions of this statement: we see that the presence of a particular topological feature in the boundary of a hyperbolic group is determined by the geometry of balls in the Cayley graph of radius bounded above by some known upper bound, and is therefore algorithmically detectable. Using these technical results one can prove that certain properties of a group can be computed from its presentation. In particular, we show that there are algorithms that, when given a presentation for a one-ended hyperbolic group, compute Bowditch’s canonical decomposition of that group and determine whether or not that group is virtually Fuchsian. The final chapter of this thesis studies the problem of detecting Cechˇ cohomological features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm that computes the Cechˇ cohomology of the boundary of a given hyperbolic group. We answer Epstein’s question in the affirmative for a restricted class of hyperbolic groups: those that are fundamental groups of graphs of free groups with cyclic edge groups. We also prove the computability of the Cechˇ cohomology of a space with some similar properties to the boundary of a hyperbolic group: Otal’s decomposition space associated to a line pattern in a free group. Declaration I hereby declare that this dissertation is the result of my own work and includes nothing that is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text Benjamin Barrett November 2018 Acknowledgements First and foremost I would like to express my deep gratitude to my supervisor, Henry Wilton, whose guidance, encouragement, positivity and patience have made these four years a rewarding and (mostly) enjoyable experience. I could not have hoped for more dedicated supervisor. I also thank the many other people in the geometric group theory community, in Cambridge and more broadly, who have taken the time to discuss my work with me, including but certainly not limited to Martin Bridson, Brian Bowditch, Daniel Groves, Panos Papasoglu, Alan Logan, John Mackay and Alex Margolis. I am grateful to the organisers of Geometric Group Theory in the South East and of the Young Geometric Group Theory conference series: these events have broadened my understanding of geometric group theory and made me a better mathematician. I thank the EPSRC for funding this PhD. I also thank Trinity College for being my home in Cambridge for seven years and Churchill College for welcoming me into its community. I thank my friends from my time as an undergraduate and graduate student at Cam- bridge, including James Munro, Fiona Hamey, David Mestel, Jack Smith, Jo Evans, Tom Berrett, Lawrence Barrott, James Kilbane, Nigel Burke and Nicolas Dupré, for their conver- sation both mathematical and (especially) non-mathematical. Living with James Munro and Fiona Hamey has made writing this thesis a more pleasant experience, and I am particularly grateful to them. I am also grateful to my friends outside the mathematical community, whose friend- ship has enriched my life and provided often-needed distraction from mathematics. I thank Cambridge Labour Party, and in particular George Owers, Dan Ratcliffe, Dan Swain and Ashley Walsh. Finally I thank my family, and in particular my parents, to whom this thesis is dedicated. It is thanks to their early encouragement and support that I have reached this point. Table of contents 1 Introduction1 1.1 Detecting connectedness properties of the boundary . .3 1.2 Computing splittings and JSJ decompositions . .4 1.3 Decomposition spaces . .8 2 Group actions on hyperbolic spaces 11 2.1 Word hyperbolicity . 11 2.1.1 The word metric and the Švarc-Milnor Lemma . 11 2.1.2 Hyperbolic geometry . 13 2.1.3 Hyperbolic groups . 16 2.2 The Gromov boundary . 17 2.2.1 Definition and topology . 18 2.2.2 The visual metric . 21 2.2.3 Local connectedness . 22 2.3 Relative hyperbolicity . 22 2.3.1 A dynamical characterisation . 23 2.3.2 The cusped space . 24 2.3.3 The Bowditch boundary . 26 3 Splittings and JSJ decompositions 27 3.1 Amalgams and HNNs . 27 3.2 More general splittings . 28 3.2.1 Graphs of groups . 28 3.2.2 Graphs of spaces . 29 3.2.3 Domination and refinement . 31 3.3 Group actions on trees . 32 3.4 Splittings over finite subgroups . 34 3.5 JSJ decompositions . 35 x Table of contents 3.5.1 Guirardel and Levitt . 36 3.5.2 Bowditch’s canonical decomposition . 37 3.5.3 Peripheral splittings . 40 3.5.4 Cut pairs in the relative setting . 41 3.6 Detecting splittings over finite groups . 42 3.6.1 The double dagger condition . 43 3.6.2 The computability of ‡ . 48 4 Cut points and cut pairs 49 4.1 Cylinders, cut points and cut pairs . 50 4.1.1 Spaces and constants . 50 4.1.2 Cylinders in the cusped space . 51 4.2 Cylinders in the thin part of the cusped space . 59 4.3 The geometry of finite balls in the cusped space . 61 4.3.1 Cut points . 62 4.3.2 Cut pairs . 62 4.3.3 Circular boundaries . 66 4.4 Generalisations . 70 5 Algorithmic consequences 71 5.1 Detecting cut points, cut pairs and circular boundaries . 72 5.1.1 Computing the constants . 72 5.1.2 Algorithms . 74 5.2 Computing splittings . 75 5.2.1 Virtually cyclic subgroups . 76 5.2.2 Enumerating the splittings . 78 5.3 Splittings of groups with circular boundary . 80 5.3.1 Groups with circular boundary . 80 5.3.2 Orbifolds . 82 5.3.3 Splittings of fundamental groups of orbifolds . 85 5.3.4 Detecting whether or not an orbifold is small . 86 5.4 Maximal splittings . 87 5.4.1 Boundaries with cut pairs . 88 5.4.2 Boundaries without cut points or pairs . 90 5.4.3 Computing a maximal splitting . 91 5.5 JSJ decompositions . 92 5.5.1 Virtually cyclic edge groups and Bowditch’s decomposition . 93 Table of contents xi 5.5.2 Z edge groups . 96 5.5.3 Zmax edge groups . 98 5.6 Generalisations . 99 6 The cohomology of decomposition spaces 101 6.1 Whitehead graphs and open covers . 102 6.2 Computing Hˇ 0(D).................................. 105 6.3 Computing Hˇ 1(D).................................. 109 6.4 The cohomology of hyperbolic graphs of free groups . 111 6.4.1 Boundaries of fundamental groups of graphs of groups . 111 6.4.2 Computing the cohomology . 113 Bibliography 119 Chapter 1 Introduction Geometric group theory has its roots in the idea that one can learn about a group from the geometric properties of its Cayley graph, or more generally the geometry of other spaces on which the group admits a nice action. Fundamental to this idea is the Švarc-Milnor Lemma, which tells us that, given a group G acting nicely on a metric space X , the large scale geometry of X depends only on G, and so large scale properties of X can be powerful invariants of the group G. This raises the following question. Question 1.0.1. On how large a scale does one need to look at a space to see its large scale features? To what extent does the geometry of a ball of large but bounded radius in the Cayley graph of a group determine the large scale geometry of that group? In this thesis we answer this question for certain large scale properties of groups satisfying an important curvature condition, called Gromov hyperbolicity. This property was introduced by Gromov [Gro87] as a coarse notion of negative curvature. Groups that act on these spaces are well behaved in very many ways, and share many nice properties with free groups, but the definition is not narrow: many groups of classical interest—such as fundamental groups of hyperbolic manifolds—are hyperbolic, as are most finitely presented groups.
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