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Elements of Asymptotic Geometry
Sergei Buyalo, Russian Academy of Sciences, St. Petersburg, Russia, and Viktor Schroeder, University of Zurich, Switzerland
A publication of the European Mathematical Society.
EMS Monographs in Mathematics
2007; 209 pp; hardcover
Volume: 3
ISBN-10: 3-03719-036-1
ISBN-13: 978-3-03719-036-4
List Price: US$78
Member Price: US$62.40
Order Code: EMSMONO/3
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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 Zürich.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and researchers working in geometry, topology, and geometric group theory.

Table of Contents

  • Hyperbolic geodesic spaces
  • The boundary at infinity
  • Busemann functions on hyperbolic spaces
  • Morphisms of hyperbolic spaces
  • Quasi-Möbius and quasi-symmetric maps
  • Hyperbolic approximation of metric spaces
  • Extension theorems
  • Embedding theorems
  • Basics of dimension theory
  • Asymptotic dimension
  • Linearly controlled metric dimension: Basic properties
  • Linearly controlled metric dimension: Applications
  • Hyperbolic dimension
  • Hyperbolic rank and subexponential corank
  • Appendix. Models of the hyperbolic space \(H^n\)
  • Bibliography
  • Index
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