Metric Spaces of Non-Positive Curvature

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 .

I. Geodesic Metric Spaces.- 1. Basic Concepts.- 2. The Model Spaces M?n.- 3. Length Spaces.- 4. Normed Spaces.- 5. Some Basic Constructions.- 6. More on the Geometry of M?n.- 7. M?-Polyhedral Complexes.- 8. Group Actions and Quasi-Isometries.- II. CAT(?) Spaces.- 1. Definitions and Characterizations of CAT(?) Spaces.- 2. Convexity and Its Consequences.- 3. Angles, Limits, Cones and Joins.- 4. The Cartan-Hadamard Theorem.- 5. M?-Polyhedral Complexes of Bounded Curvature.- 6. Isometries of CAT(0) Spaces.- 7. The Flat Torus Theorem.- 8. The Boundary at Infinity of a CAT(0) Space.- 9. The Tits Metric and Visibility Spaces.- 10. Symmetric Spaces.- 11. Gluing Constructions.- 12. Simple Complexes of Groups.- III. Aspects of the Geometry of Group Actions.- H. ?-Hyperbolic Spaces.- ?. Non-Positive Curvature and Group Theory.- C. Complexes of Groups.- G. Groupoids of local Isometries.- References.