Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at infinity. In the first part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the basic theory of Gromov hyperbolic spaces. These spaces have been studied extensively in the last twenty years and have found applications in group theory, geometric topology, Kleinian groups, as well as dynamics and rigidity theory. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces are considered. It turns out that the boundary at infinity approach is not appropriate in the general case, but dimension theory proves useful for finding interesting results and applications. The text leads concisely to some central aspects of the theory. Each chapter concludes with a separate section containing supplementary results and bibliographical notes. Here the theory is also illustrated with numerous examples as well as relations to the neighboring fields of comparison geometry and geometric group theory. The book is based on lectures the authors presented at the Steklov Institute in St. Petersburg and the University of Z??rich. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Sergei Buyalo, Viktor Schroeder
Year: 2007
Language: English
Pages: 209
Contents......Page 8
Preface......Page 12
Geodesic metric spaces......Page 14
Hyperbolic geodesic spaces......Page 15
Stability of geodesics......Page 16
Supplementary results and remarks......Page 19
-inequality and hyperbolic spaces......Page 22
The boundary at infinity of hyperbolic spaces......Page 25
Local self-similarity of the boundary......Page 29
Supplementary results and remarks......Page 32
Busemann functions......Page 36
Gromov products based at infinity......Page 39
Visual metrics based at infinity......Page 42
Supplementary results and remarks......Page 43
4 Morphisms of hyperbolic spaces......Page 48
Morphisms of metric spaces and hyperbolicity......Page 49
Cross-difference triples and cross-differences......Page 52
PQ-isometric maps......Page 54
Quasi-isometric maps of hyperbolic geodesic spaces......Page 55
Supplementary results and remarks......Page 57
Cross-ratios......Page 62
Quasi-Möbius and quasi-symmetric maps......Page 63
Supplementary results and remarks......Page 69
Summary......Page 76
Construction......Page 82
Geodesics in a hyperbolic approximation......Page 83
The boundary at infinity of a hyperbolic approximation......Page 87
Supplementary results and remarks......Page 90
Extension theorem for bilipschitz maps......Page 94
Extension theorem for quasi-symmetric maps......Page 97
Extension theorem for quasi-Möbius maps......Page 100
Supplementary results and remarks......Page 108
Assouad embedding theorem......Page 110
Bonk–Schramm embedding theorem......Page 113
Supplementary results and remarks......Page 115
Various dimensions......Page 120
Constructions......Page 124
P-dimensions......Page 130
The monotonicity theorem......Page 134
The saturation of families......Page 135
The finite union theorem......Page 136
Sperner lemma......Page 137
Supplementary results and remarks......Page 139
Estimates from below......Page 142
Estimates from above......Page 143
Embedding of H^2 into a product of two trees......Page 145
Supplementary results and remarks......Page 147
11 Linearly controlled metric dimension: Basic properties......Page 150
Separated sequences of colored coverings......Page 151
Quasi-symmetry invariance of l-dim......Page 154
Supplementary results and remarks......Page 158
Embedding into the product of trees......Page 160
-dimension of locally self-similar spaces......Page 167
Applications to hyperbolic spaces......Page 169
Supplementary results and remarks......Page 170
Large scale doubling sets......Page 172
Definition of the hyperbolic dimension......Page 173
Hyperbolic dimension of hyperbolic spaces......Page 175
Applications to nonembedding results......Page 177
Supplementary results and remarks......Page 179
Hyperbolic rank......Page 180
Subexponential corank......Page 182
Subexponential corank versus hyperbolic dimension......Page 188
Supplementary results and remarks......Page 190
The pseudo-spherical model......Page 194
The unit disc model......Page 195
The upper half-plane model......Page 196
The solvable group model......Page 198
Generalizations to an arbitrary dimension......Page 199
Möbius transformations......Page 200
Cross-ratio......Page 201
Bibliography......Page 206
Index......Page 210
