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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 .

Espaces métriques --- Geometry [Differential ] --- Géométrie différentielle --- Meetkunde [Differentiaal] --- Metric spaces --- Ruimten [Metrische ] --- Geometry, Differential --- 514.764.2 --- Spaces, Metric --- Generalized spaces --- Set theory --- Topology --- Differential geometry --- Riemannian and pseudo-Riemannian spaces --- Geometry, Differential. --- Metric spaces. --- 514.764.2 Riemannian and pseudo-Riemannian spaces --- Geometric group theory --- Groupes, Théorie géométrique des --- Topology. --- Manifolds (Mathematics). --- Complex manifolds. --- Group theory. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Analytic spaces --- Manifolds (Mathematics) --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Algebras, Linear --- Groupes, Théorie géométrique des. --- Géometrie différentielle globale --- Géometrie différentielle globale --- Topologie differentielle