## 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

Geodesic Metric Spaces | 1 |

The Model Spaces 11m 15 2 The Model Spaces | 15 |

Length Spaces 32 3 Length Spaces | 32 |

Normed Spaces 17 1 Normed Spaces | 47 |

Some Basic Constructions 56 5 Some Basic Constructions | 56 |

More on the Geometry of M | 81 |

MKPolyhedral Complexes | 97 |

Group Actions and QuasiIsometries | 131 |

Simple Complexes of Groups | 367 |

Aspects of the Geometry of Group Actions | 397 |

T NonPositive Curvature and Group Theory 138 | 438 |

Amalgamating Groups of Isometries | 496 |

FiniteSheeted Coverings and Residual Finiteness | 511 |

Complexes of Groups | 519 |

Complexes of Groups | 534 |

The Fundamental Group of a Complex of Groups | 546 |

CATK Spaces | 157 |

Angles Limits Cones and Joins | 184 |

The CartanHadamard Theorem | 193 |

Isometries of CAT0 Spaces | 228 |

The Flat Torus Theorem 217 | 244 |

The Boundary at Infinity of a CAT0 Space | 260 |

The Tits Metric and Visibility Spaces | 277 |

Symmetric Spaces | 299 |

Gluing Constructions | 347 |

Local Developments of a Complex of Groups | 555 |

Coverings of Complexes of Groups | 566 |

G Groupoids of local Isometries | 584 |

Étale Groupoids Homomorphisms and Equivalences | 594 |

The Fundamental Group and Coverings of Étale Groupoids | 604 |

Proof of the Main Theorem | 613 |

620 | |

637 | |

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