Beyond Hyperbolicity London Mathematical Society Lecture Note Series

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.

Preface; Part I. Lectures: 1. Notes on coarse median spaces Brian H. Bowditch; 2. Semihyperbolicity Martin R. Bridson; 3. Acylindrically hyperbolic groups Benjamin Barrett; Part II. Expository Articles: 4. A survey on Morse boundaries and stability Matthew Cordes; 5. What is a hierarchically hyperbolic space? Alessandro Sisto; Part III. Research Articles: 6. A counterexample to questions about boundaries, stability, and commensurability Jason Behrstock; 7. A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes Indira Chatterji and Alexandre Martin; 8. Immutability is not uniformly decidable in hyperbolic groups Daniel Groves and Henry Wilton; 9. Sphere systems, standard form, and cores of products of trees Francesca Iezzi; 10. Uniform quasiconvexity of the disc graphs in the curve graphs Kate M. Vokes.

Mark Hagen is a Lecturer in Mathematics at the University of Bristol. His interests lie in geometric group theory, including in particular cubical/median geometry, mapping class groups, and their coarse-geometric generalisations.
Richard Webb is an EPSRC Postdoctoral Fellow at the University of Cambridge and a Stokes Research Fellow at Pembroke College. He investigates the algebra and geometry of the mapping class group and its relatives, often using techniques and inspiration drawn from geometric group theory.
Henry Wilton is a Reader in Pure Mathematics at the University of Cambridge and a Fellow of Trinity College. He works in the fields of geometric group theory and low-dimensional topology. His interests include the subgroup structure of hyperbolic groups, questions of profinite rigidity, decision problems, and properties of 3-manifold groups.