Metric Spaces of Non-Positive Curvature

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Springer Science & Business Media, Oct 20, 2011 - Mathematics - 643 pages
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 .
 

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Contents

Geodesic Metric Spaces
1
The Model Spaces 11m 15 2 The Model Spaces
15
Length Spaces 32 3 Length Spaces
32
Normed Spaces 17 1 Normed Spaces
47
Some Basic Constructions 56 5 Some Basic Constructions
56
More on the Geometry of M
81
MKPolyhedral Complexes
97
Group Actions and QuasiIsometries
131
Simple Complexes of Groups
367
Aspects of the Geometry of Group Actions
397
T NonPositive Curvature and Group Theory 138
438
Amalgamating Groups of Isometries
496
FiniteSheeted Coverings and Residual Finiteness
511
Complexes of Groups
519
Complexes of Groups
534
The Fundamental Group of a Complex of Groups
546

CATK Spaces
157
Angles Limits Cones and Joins
184
The CartanHadamard Theorem
193
Isometries of CAT0 Spaces
228
The Flat Torus Theorem 217
244
The Boundary at Infinity of a CAT0 Space
260
The Tits Metric and Visibility Spaces
277
Symmetric Spaces
299
Gluing Constructions
347
Local Developments of a Complex of Groups
555
Coverings of Complexes of Groups
566
G Groupoids of local Isometries
584
Étale Groupoids Homomorphisms and Equivalences
594
The Fundamental Group and Coverings of Étale Groupoids
604
Proof of the Main Theorem
613
References
620
Index
637

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