Beyond Hyperbolicity

doi:10.1017/9781108559065

Since the notion was introduced by Gromov in the 1980s, hyperbolicity of groups and spaces has played a significant role in geometric group theory; hyperbolic groups have good geometric properties that allow us to prove strong results. However, many classes of interest in our exploration of the universe of finitely generated groups contain examples that are not hyperbolic. Thus we wish to go 'beyond hyperbolicity' to find good generalisations that nevertheless permit similarly strong results. This book is the ideal resource for researchers wishing to contribute to this rich and active field. The first two parts are devoted to mini-courses and expository articles on coarse median spaces, semihyperbolicity, acylindrical hyperbolicity, Morse boundaries, and hierarchical hyperbolicity. These serve as an introduction for students and a reference for experts. The topics of the surveys (and more) re-appear in the research articles that make up Part III, presenting the latest results beyond hyperbolicity.

‘The articles in this collection take the reader on a journey from foundational examples and definitions to state-of-the-art theorems and actively researched open problems … Students and those wanting to enter these fields of specialization will want to return to the surveys and expositions for inspiration as they engage with more specialized literature.’

Robert Bell Source: MAA Reviews

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Beyond Hyperbolicity

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

Part One - LECTURES
pp 1-2

1 - Notes on coarse median spaces
pp 3-24

2 - Semihyperbolicity
pp 25-64

3 - Acylindrically hyperbolic groups
pp 65-80

Part Two - Expository Articles
pp 81-82

4 - A survey on Morse boundaries and stability
pp 83-116

5 - What is a hierarchically hyperbolic space?
pp 117-148

Part Three - Research Articles
pp 149-150

6 - A counterexample to questions about boundaries, stability, and commensurability
pp 151-159

7 - A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes
pp 160-178

8 - Immutability is not uniformly decidable in hyperbolic groups
pp 179-185

9 - Sphere systems, standard form, and cores of products of trees
pp 186-222

10 - Uniform quasiconvexity of the disc graphs in the curve graphs
pp 223-231

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