## Metric Spaces of Non-Positive CurvatureThe purpose of this book is to describe the global properties of complete simply connected spaces that are non-positively curved in the sense of A. D. Alexandrov and to examine the structure of groups that act properly on such spaces by isometries. Thus the central objects of study are metric spaces in which every pair of points can be joined by an arc isometric to a compact interval of the real line and in which every triangle satisfies the CAT(O) inequality. This inequality encapsulates the concept of non-positive curvature in Riemannian geometry and allows one to reflect the same concept faithfully in a much wider setting - that of geodesic metric spaces. Because the CAT(O) condition captures the essence of non-positive curvature so well, spaces that satisfy this condition display many of the elegant features inherent in the geometry of non-positively curved manifolds. There is therefore a great deal to be said about the global structure of CAT(O) spaces, and also about the structure of groups that act on them by isometries - such is the theme of this book. 1 The origins of our study lie in the fundamental work of A. D. Alexandrov . |

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### Contents

ount A In O ovſ11soduo N | 1 |

Spherical Joins | 63 |

Limits of Metric Spaces | 70 |

Ultralimits and Asymptotic Cones | 77 |

ou L | 88 |

MºPolyhedral Complexes | 97 |

soonds LVO Jo Suſtuſ lulodf | 98 |

Group Actions and QuasiIsometries | 132 |

Further Properties of Hyperbolic Groups | 460 |

Semihyperbolic Groups | 471 |

Subgroups of Cocompact Groups of Isometries | 481 |

Amalgamating Groups of Isometries | 496 |

of Doubles | 506 |

Complexes of Groups | 520 |

Complexes of Groups | 534 |

The Fundamental Group of a Complex of Groups | 546 |

CATk Spaces | 157 |

SL los popunog º Jo Ouluo | 177 |

8 | 260 |

Fundamental Groups and Coverings | 314 |

Aspects of the Geometry of Group Actions | 397 |

NonPositive Curvature and Group Theory | 438 |

Local Developments of a Complex of Groups | 555 |

Coverings of Complexes of Groups | 566 |

Groupoids of local Isometries | 584 |

Etale Groupoids Homomorphisms and Equivalences 59 4 | 594 |

The Fundamental Group and Coverings of Étale Groupoids | 604 |

Proof of the Main Theorem | 616 |

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### Common terms and phrases

5ululoſ alou aloul alouw asoul aspo barycentric barycentric subdivision CAT(k Cayley graph compact comparison triangle complex of groups convex defined denote equivalence relation Euclidean finite fundamental group geodesic segment geodesic space HNN extension homeomorphism hyperbolic hyperbolic group hyperplane isometry isomorphic Joold Lemma length lonpold loſ lºul lulod metric space morphism oldus olloquod ºrds ºut ovoid posolo poulop Proof pull put ºn quasi-isometry Riemannian metric run uons scwol semihyperbolic simplicial ſlºw slul soonds sºul ſpool subgroup subset subspace swollo Theorem tº put tº St tº uo tº uſ triangle tuouſ unlºw unlu unns unpo unsual uonor uoul uoun uouſ vertex vertices wouls

### Popular passages

Page 630 - P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Ann.

### References to this book

A Course in Metric Geometry Dmitri Burago,I͡Uriĭ Dmitrievich Burago,Sergeĭ Ivanov No preview available - 2001 |