Metric Spaces of Non-Positive Curvature (Grundlehren der mathematischen Wissenschaften (319))

The 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 . Front Matter....Pages I-XXI Front Matter....Pages 1-1 Basic Concepts....Pages 2-14 Length Spaces....Pages 15-31 Normed Spaces....Pages 32-46 Some Basic Constructions....Pages 47-55 M κ —Polyhedral Complexes....Pages 56-80 Group Actions and Quasi-Isometries....Pages 81-96 Front Matter....Pages 97-130 Definitions and Characterizations of CAT( κ ) Spaces....Pages 131-156 Convexity and its Consequences....Pages 157-157 Angles, Limits, Cones and Joins....Pages 158-174 The Cartan-Hadamard Theorem....Pages 175-183 M к -Polyhedral Complexes of Bounded Curvature....Pages 184-192 Isometries of CAT(0) Spaces....Pages 193-204 The Flat Torus Theorem....Pages 205-227 The Boundary at Infinity of a CAT(0) Space....Pages 228-243 The Tits Metric and Visibility Spaces....Pages 244-259 Symmetric Spaces....Pages 260-276 Gluing Constructions....Pages 277-298 Simple Complexes of Groups....Pages 299-346 Front Matter....Pages 347-366 δ -Hyperbolic Spaces and Area....Pages 367-396 Non-Positive Curvature and Group Theory....Pages 397-397 Complexes of Groups....Pages 398-437 Groupoids of Local Isometries....Pages 438-518 Back Matter....Pages 519-583 ....Pages 584-619 The purpose of this book is to describe the global properties of complete simplyƯ 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 This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I is an introduction to the geometry of geodesic spaces. In Part II the basic theory of spaces with upper curvature bounds is developed. More specialized topics, such as complexes of groups, are covered in Part III. The book is divided into three parts, each part is divided into chapters and the chapters have various subheadings. The chapters in Part III are longer and for ease of reference are divided into numbered sections.

A description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.

The fundamental concept with which we shall be concerned throughout this book is that of distance.