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El. knyga: Beyond Hyperbolicity

Edited by (University of Bristol), Edited by (University of Cambridge), Edited by (University of Cambridge)
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This book focuses on generalisations of Gromov hyperbolicity in geometric group theory. Five self-contained expository articles introduce topics 'beyond hyperbolicity': these can be used as an introduction for students or as a reference for experts. The final part contains research articles on the latest results in this rich and active field.

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.

Recenzijos

'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, MAA Reviews

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Contains expository articles and research papers in geometric group theory focusing on generalisations of Gromov hyperbolicity.
Preface ix
PART ONE LECTURES
1(80)
1 Notes on coarse median spaces
3(22)
B.H. Bowditch
1.1 Introduction
3(1)
1.2 Quasi-isometry invariants
4(4)
1.3 Medians
8(3)
1.4 Coarse median spaces
11(2)
1.5 Surfaces
13(5)
1.6 Asymptotic cones
18(7)
2 Semihyperbolicity
25(40)
M.R. Bridson
2.1 The universe of finitely presented groups
27(3)
2.2 Some key features of hyperbolic groups
30(7)
2.3 Some properties of CAT(0) groups
37(1)
2.4 Combings and semihyperbolicity
38(8)
2.5 Languages and the complexity of normal forms
46(6)
2.6 Examples
52(4)
2.7 Algorithmic construction of classifying spaces
56(1)
2.8 Cubulated groups and systolic groups
57(1)
2.9 Subgroups
58(1)
2.10 Containments
59(6)
3 Acylindrically hyperbolic groups
65(16)
B. Barrett
3.1 Acylindrically hyperbolic groups
65(4)
3.2 Small-cancellation theory
69(3)
3.3 Dehn surgery
72(3)
3.4 The extension problem
75(3)
3.5 Acylindrically hyperbolic structures
78(3)
PART TWO EXPOSITORY ARTICLES
81(68)
4 A survey on Morse boundaries and stability
83(34)
M. Cordes
4.1 Generalizing hyperbolicity
83(2)
4.2 Contracting and Morse boundaries
85(10)
4.3 (Metric) Morse boundary and stability
95(11)
4.4 Stable subgroups
106(4)
4.5 A metrisable topology on the Morse boundary
110(7)
5 What is a hierarchically hyperbolic space?
117(32)
A. Sisto
5.1 Heuristic discussion
120(9)
5.2 Technical discussion
129(20)
PART THREE RESEARCH ARTICLES
149
6 A counterexample to questions about boundaries, stability, and commensurability
151(9)
J. Behrstock
6.1 The construction
152(1)
6.2 Properties
153(2)
6.3 Applications
155(2)
6.4 Further questions
157(3)
7 A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes
160(19)
I. Chatterji
A. Martin
7.1 Introduction
160(4)
7.2 Uber-contractions and acylindrical hyperbolicity
164(1)
7.3 Uber-separated hyperplanes and the proof of Theorem 7.1.1
165(5)
7.4 Proof of Theorem 7.1.5
170(3)
7.5 Artin groups of type FC
173(6)
8 Immutability is not uniformly decidable in hyperbolic groups
179(7)
D. Groves
H. Wilton
9 Sphere systems, standard form, and cores of products of trees
186(37)
F. Iezzi
9.1 Introduction
186(3)
9.2 Spheres, partitions and intersections
189(2)
9.3 Standard form for sphere systems, piece decomposition and dual square complexes
191(6)
9.4 The core of two trees
197(10)
9.5 The inverse construction
207(10)
9.6 Consequences and applications
217(6)
10 Uniform quasiconvexity of the disc graphs in the curve graphs
223
K.M. Vokes
10.1 Introduction
223(2)
10.2 Preliminaries
225(1)
10.3 Exceptional cases
226(1)
10.4 Proof of the main result
226
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.
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