Large Scale Geometry

✍ Scribed by Piotr W. Nowak, Guoliang Yu

• Publisher: European Mathematical Society
• Year: 2012
• Tongue: English
• Leaves: 203
• Series: EMS Textbooks in Mathematics
• Category: Library

✦ Synopsis

Large scale geometry is the study of geometric objects viewed from a great distance. The idea of large scale geometry can be traced back to Mostow's work on rigidity and the work of varc, Milnor, and Wolf on growth of groups. In the last decades, large scale geometry has found important applications in group theory, topology, geometry, higher index theory, computer science, and large data analysis. This book provides a friendly approach to the basic theory of this exciting and fast growing subject and offers a glimpse of its applications to topology, geometry, and higher index theory. The authors have made a conscientious effort to make the book accessible to advanced undergraduate students, graduate students, and non-experts. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
Notation and conventions......Page 13
1.1 Metric spaces......Page 15
1.2 Groups as metric spaces......Page 20
1.3 Quasi-isometries......Page 23
1.4 Coarse equivalences......Page 29
1.5 Hyperbolic spaces......Page 33
Exercises......Page 38
Notes and remarks......Page 39
2.1 Topological dimension......Page 40
2.2 Asymptotic dimension......Page 41
2.3 Dimension of hyperbolic groups......Page 44
2.4 Upper bounds for asymptotic dimension......Page 46
2.5 Asymptotic dimension of solvable groups......Page 50
2.6 Groups with infinite asymptotic dimension......Page 52
2.7 Decomposition complexity......Page 53
2.8 Invariance and permanence......Page 56
2.9 Groups with finite decomposition complexity......Page 59
Notes and remarks......Page 61
3.1 Følner conditions......Page 63
3.2 The Hulanicki–Reiter condition......Page 69
3.3 Invariant means......Page 72
Notes and remarks......Page 75
4.1 Definition and basic properties......Page 77
4.2 The Higson–Roe condition......Page 80
4.3 Finite asymptotic dimension implies property A......Page 84
4.4 Property A and residually finite groups......Page 87
4.5 Locally finite examples......Page 92
Notes and remarks......Page 95
5.1 Coarse embeddings......Page 97
5.2 Embeddability into Hilbert spaces......Page 98
5.3 Examples of embeddable spaces without property A......Page 103
5.4 Convexity and reflexivity......Page 105
5.5 Coarse embeddings and finite subsets......Page 110
5.6 Expanders......Page 112
5.7 A geometric characterization of non-embeddability......Page 115
5.8 Compression of coarse embeddings......Page 122
5.9 Compression >12 implies property A......Page 124
Exercises......Page 130
Notes and remarks......Page 131
6.1 Affine isometric actions......Page 133
6.2 Metrically proper actions and a-T-menability......Page 135
6.3 Actions on _p-spaces and reflexive Banach spaces......Page 139
6.4 Kazhdan's property (T)......Page 142
6.5 Fixed points and Kazhdan's property (T)......Page 144
6.6 Construction of expanders......Page 146
6.7 Property (T) and spectral conditions......Page 148
Exercises......Page 154
Notes and remarks......Page 155
7.1 Coarse locally finite homology......Page 157
7.2 Uniformly finite homology......Page 158
7.3 Eilenberg swindles and Ponzi schemes......Page 163
7.4 Aperiodic tiles and non-amenable spaces......Page 168
7.5 Coarsening homology theories......Page 174
7.6 The coarsening homomorphism......Page 176
Notes and remarks......Page 179
8.1 Topological rigidity......Page 180
8.2 Geometric rigidity......Page 181
8.3 Index theory......Page 183
References......Page 187
Index......Page 201