# Elements of Asymptotic Geometry

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*Sergei Buyalo; Viktor Schroeder*

A publication of the European Mathematical Society

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.

#### Readership

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