## Elements of Asymptotic GeometryAsymptotic 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. |

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### 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 |

193 | |

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asdim assume asymptotic dimension balls B(v barycentric subdivision base point bilipschitz boundary at infinity bounded metric space Busemann function b e consider constant contains control functions converges to infinity Corollary covering tí cr(Q cross-ratio define definition diam distance distinct points edges estimate finite follows Furthermore Gromov hyperbolic Gromov product Hadamard manifold Hence homeomorphism hyperbolic approximation hyperbolic dimension hyperbolic geodesic spaces hyperbolic group hyperbolic space inequality isometric l)-colored large scale doubling Lebesgue number Lemma linearly controlled mesh(tí metric space metric trees Möbius Möbius transformations multiplicative obtain open covering parameter PQ-isometric map proof of Theorem Proposition quadruple Q quasi-isometric embedding quasi-isometric map quasi-metric space quasi-Möbius map quasi-symmetric radial respect sectional curvature sequence simplex simplicial strongly PQ-isometric subset topological dimension triangle triangle inequality uniformly perfect vertices visual metric