Elements of Asymptotic Geometry

Front Cover
European Mathematical Society, 2007 - Algebraic topology - 200 pages
0 Reviews
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 Zurich.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

The boundary at infinity
9
Busemann functions on hyperbolic spaces
23
Morphisms of hyperbolic spaces
35
QuasiMöbius and quasisymmetric maps
49
Hyperbolic approximation of metric spaces
69
Extension theorems
81
Embedding theorems
97
Basics of dimension theory
107
Asymptotic dimension
129
Basic properties
137
Applications
147
Hyperbolic dimension
159
Hyperbolic rank and subexponential corank
167
Appendix Models of the hyperbolic space H
181
Bibliography
193
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information