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.

Inhalt

PART I: Geodesic Metric Spaces: BASIC CONCEPTS. THE MODEL SPACES Mnk.- LENGTH SPACES.- NORMED SPACES.- SOME BASIC CONSTRUCTIONS.- MORE ON THE GEOMETRY OF $M Ök$^.- $M Ök$-POLYHEDRAL COMPLEXES.- APPENDIX 7A: Metrizing abstract simplicial complexes.- GROUP ACTIONS AND QUASI-ISOMETRIES.- APPENDIX 8A: Combinatorial 2-complexes.- Part II: CAT($Ökappa$) Spaces.- DEFINITIONS AND CHARACTERIZATIONS OF CAT($Ökappa$) SPACES.- CONVEXITY AND ITS CONSEQUENCES.- ANGLES, LIMITS, CONES AND JOINS.- THE CARTAN-HADAMARD THEOREM.- $M Ök$-POLYHEDRAL COMPLEXES.- ISOMETRIES OF CAT(0) SPACES.- THE FLAT TORUS THEOREM.- THE BOUNDARY AT INFINITY OF A CAT(0) SPACE.- THE TITS METRIC AND VISIBILITY SPACES.- SYMMETRIC SPACES.- APPENDIX 10A: Spherical and Euclidean buildings.- CONSTRUCTIONS INVOLVING GLUING.- SIMPLE COMPLEXES OF GROUPS.- Part III: Topics in non-positive curvature.- $Ödelta$-HYPERBOLIC SPACES.- $ÖGamma$: NON-POSITIVE CURVATURE AND GROUP THEORY.-.$ÖCal C$: COMPLEXES OF GROUPS.- .$ÖCal G$: GROUPOIDS OF LOCAL ISOMETRIES.