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Morse, Contracting, and Strongly Contracting Sets with Applications to Boundaries and Growth of Groups

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Morse, Contracting, and Strongly Contracting Sets with Applications to Boundaries and Growth of Groups Morse, contracting, and strongly contracting sets with applications to boundaries and growth of groups Christopher H. Cashen, PhD Faculty of Mathematics University of Vienna 1090 Vienna,Austria E-mail address: [email protected] URL: http://www.mat.univie.ac.at/~cashen Key words and phrases. Morse, contracting, strongly contracting, graphical small cancellation, group action, boundary, growth tightness, cogrowth The author was supported by the Austrian Science Fund (FWF): P 30487-N35 during the writing of this thesis. Support for the individual papers can be found in the respective chapters. Abstract. We investigate several quantitative generalizations of the notion of quasiconvex subsets of (Gromov) hyperbolic spaces to arbitrary geodesic metric spaces. Some of these, such as the Morse property, strong contraction, and superlinear divergence, had been studied before in more specialized contexts, and some, such as contraction, we introduce for the first time. In general, we prove that quasiconvexity is the weakest of the properties, strong contraction is the strongest, and all of the others are equivalent. However, in hyperbolic spaces all are equivalent, and we prove that in CAT(0) spaces all except quasiconvexity are equivalent. Despite the fact that many of these properties are equivalent, they are useful for different purposes. For instance, it is easy to see that the Morse property is quasiisometry invariant, but the contraction property gives good control over the divagation behavior of geodesic rays with a common basepoint. We exploit this control to define a boundary for arbitrary finitely generated groups that shares some properties of the boundary of a hyperbolic group. Our boundary is a metrizable topological space that is invariant under quasiisometries of the group, and the group acts on it with simple dynamics. We investigate the geometry of infinitely presented graphical small cancellation groups. Such groups include the so-called ‘Gromov Monsters’, which were introduced as a source of counter-examples to the Baum-Connes conjecture. We give a local-to-global characterization of contracting geodesics in these groups, which we think of as defining ‘hyperbolic directions’. Our characterization depends on a beautiful interplay between combinatorial and geometric versions of negative curvature. The result shows that the geometry of these groups is reminiscent of the geometry of relatively hyperbolic groups in the sense that there are certain well-defined non-hyperbolic regions, and geodesics that avoid these regions behave like hyperbolic geodesics. However, the groups are in general not relatively hyperbolic. Armed with our understanding of geodesics in graphical small cancellation groups, we construct examples of the wide varieties of contraction behaviors that occur: we show that every degree of contraction can be achieved by a periodic geodesic in some finitely generated group, that there are groups in which every element has a strongly contracting axis even though the group is not hyperbolic, and that there are examples of finitely generated groups in which the existence of a strongly contracting axis for a given element depends on the choice of generating set for the group. Since the Morse property is invariant under quasiisometries, we can say that a subgroup of a finitely generated group is Morse (or equivalently, contracting, divergent,...) if it has this property as a subset in some/any Cayley graph of the group. However, we could also let the group act on some other geodesic metric space and ask which elements have a contracting/Morse/strongly contracting axis for that particular action. In particular, we explore actions that are not cocompact. To preserve a connection with the geometry of the group, we require the action to be metrically proper. We also introduce a condition known as ‘complementary growth gap’ that says that there is an orbit of the group in the space that, while metrically distorted, is not too badly distorted from a growth-theoretic point of view. Our condition generalizes the ‘parabolic growth gap’ condition for Kleinian groups, and includes additional examples such as the action of the mapping class group of a hyperbolic surface on its Teichmüller space. We prove growth and cogrowth results for the orbit pseudometric induced on the group by such an action under the hypothesis the group has one element that acts with a strongly contracting axis. Our results generalize results that were known for word metrics on hyperbolic groups to far more general situations. The generality of our results is even more striking considering that there are contemporaneous papers that achieve similar results only for actions on hyperbolic spaces. Contents Preface 5 I. Plan of the thesis5 II. Synopsis of the Publications5 III. Curriculum Vitae 11 Survey of the topic and the impacts of this thesis 15 1. Introduction 15 2. Preliminaries 17 3. The Morse property and related properties 17 4. Application: The geometry of graphical small cancellation groups 28 5. Generalizing hyperbolic boundaries 30 6. Stronger properties 33 7. Application: Growth of groups 35 8. Appendix 38 References 47 Paper A. Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction 51 1. Introduction 51 2. Preliminaries 55 3. Examples of contraction 56 4. The Morse property 58 5. Divergence 62 6. Robustness 66 7. Geodesic image theorem 67 8. Further applications 69 References 71 Paper B. Morse subsets of CAT(0) spaces are strongly contracting 73 References 75 Paper C. Negative curvature in graphical small cancellation groups 77 1. Introduction 77 2. Preliminaries 83 3. Classification of quadrangles 85 0 4. Contraction in Gr (1=6)–groups 95 5. Strongly contracting elements 103 6. Hyperbolically embedded subgroups 106 References 110 Paper D. Growth Tight Actions 113 0. Introduction 113 1. Preliminaries 118 2. Contraction and Constriction 120 3. Abundance of Strongly Contracting Elements 126 4. A Minimal Section 129 5. Embedding a Free Product Set 130 3 4 CONTENTS 6. Growth Gap 130 7. Growth of Conjugacy Classes 132 8. Actions on Relatively Hyperbolic Spaces 133 9. Rank 1 Actions on CAT(0) Spaces 137 10. Mapping Class Groups 138 11. Snowflake Groups 139 References 142 Paper E. Growth Tight Actions of Product Groups 147 1. Introduction 147 2. Preliminaries 149 3. Elements that are Strongly Contracting in each Coordinate 151 4. Elements without Long, Positive Projections 153 5. Proof of the Main Theorem 155 References 157 Paper F. Cogrowth for group actions with strongly contracting elements 159 1. Introduction 159 2. Preliminaries 161 3. Embedding a tree’s worth of copies of [c] 163 4. Questions 170 References 171 Paper G. Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology 173 1. Introduction 173 2. The contracting boundary and Gromov product topology 173 3. Pathological Examples 175 References 176 Paper H. A metrizable topology on the contracting boundary of a group 177 1. Introduction 177 2. Preliminaries 179 3. Contraction 181 4. Contraction and Quasi-geodesics 184 5. The contracting boundary and the topology of fellow-travelling quasi-geodesics 189 6. Quasi-isometry invariance 193 7. Comparison to other topologies 198 8. Metrizability for group boundaries 204 9. Dynamics 206 10. Compactness 208 References 212 Preface I. Plan of the thesis This is a cumulative Habilitation thesis on the topic of generalizations of the geometric notions of convexity and hyperbolicity and their applications to the study of infinite discrete groups. The thesis includes eight papers on this topic. This Preface contains a brief Synopsis of each paper including its abstract, publication details, and a list of works that cite it, as well as a CV. The following chapter is a Survey of the research topic that gives an exposition of our results, places them into context, and demonstrates the impact of our research on its field. Each of the remaining eight chapters contains one of the papers. Each of the papers included in this thesis has undergone successful peer review and is either already published or accepted and in press. In all cases the version of the paper presented here is the final author version, post-refereeing, subjected to reformatting in a consistent style. The final version of record of each paper can be found at the listed address on the website of its publisher, and may have formatting and editorial changes. II. Synopsis of the Publications A. Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction, Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Documenta Mathematica 22 (2017), 1193–1224. https://www.math.uni-bielefeld.de/documenta/vol-22/36.html We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments. The authors were equal partners in the conception, execution, and writing of this paper. Works citing PaperA. [1] Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham. Largest acylindrical actions and stability in hierarchically hyperbolic groups. preprint. 2017. arXiv: 1705. 06219v2. [2] Tarik Aougab, Matthew Gentry Durham, and Samuel J. Taylor. Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups. J. Lond. Math. Soc. 96.3 (2017), pp. 565–583. [3] Arthur Bartels and Mladen Bestvina. The Farrell-Jones conjecture for mapping class groups. Invent. Math. 215.2 (2019), pp. 651–712. [4] Jonas Beyrer and Elia Fioravanti. Cross ratios and cubulations of hyperbolic groups. preprint. arXiv: 1810.08087. [5] Christopher H. Cashen. Morse subsets of CAT(0) spaces are strongly contracting. Geom. Dedicata (in press). [6] Matthew Cordes. A survey on Morse boundaries and stability. Beyond Hyperbolicity. London Mathematical Society Lecture Note Series.
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