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 . |
Contents
| 1 | |
| 15 | |
12 | 37 |
Manifolds of Constant Curvature | 45 |
Normed Spaces | 53 |
More on the Geometry of M | 81 |
56 | 120 |
Group Actions and QuasiIsometries | 131 |
Aspects of the Geometry of Group Actions | 397 |
The Gromov Boundary of a 8Hyperbolic Space | 427 |
г NonPositive Curvature and Group Theory | 438 |
Hyperbolic Groups and Their Algorithmic Properties | 448 |
Further Properties of Hyperbolic Groups | 459 |
Subgroups of Cocompact Groups of Isometries | 481 |
Amalgamating Groups of Isometries | 496 |
FiniteSheeted Coverings and Residual Finiteness | 511 |
CATK Spaces | 157 |
Convexity and Its Consequences | 175 |
Angles Limits Cones and Joins | 184 |
The CartanHadamard Theorem | 193 |
Isometries of CAT0 Spaces | 228 |
The Flat Torus Theorem | 244 |
The Boundary at Infinity of a CAT0 Space | 260 |
The Tits Metric and Visibility Spaces | 277 |
Symmetric Spaces | 299 |
Gluing Constructions | 347 |
Simple Complexes of Groups | 367 |
Complexes of Groups | 519 |
Complexes of Groups | 534 |
The Fundamental Group of a Complex of Groups | 546 |
Local Developments of a Complex of Groups | 555 |
Coverings of Complexes of Groups | 566 |
G Groupoids of local Isometries | 584 |
The Fundamental Group and Coverings of Étale Groupoids | 604 |
Proof of the Main Theorem | 613 |
References | 620 |
| 637 | |
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Common terms and phrases
8-hyperbolic action acts properly angle associated ball barycentric subdivision bijection c₁ CAT(K Cayley graph cocompactly by isometries compact comparison triangle complex of groups consider construction contained converges convex hull defined definition denote edge elements equivalence relation Euclidean exists finitely presented fixed fundamental group geodesic ray geodesic segment geodesic space geodesic triangle geometry given GL(n graph group G groupoid hence homeomorphism homomorphism homotopy hyperbolic groups hyperplane induced inequality integer intersection isomorphic joining Lemma length metric length space Lk(x local isometry m-string manifold metric space morphism non-positively curved orbifold path polyhedral complexes Proof properly and cocompactly Proposition prove pseudometric quasi-isometry quotient resp restriction Riemannian manifold Riemannian metric scwol semihyperbolic sequence Shapes(K simplex simplicial complex simply connected subgroup subset subspace Theorem topology triangle inequality unique geodesic vector vertex vertices X₁ y₁