Metric Spaces of Non-Positive Curvature

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

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Contents

ount A In O ovſ11soduo N
1
Spherical Joins
63
Limits of Metric Spaces
70
Ultralimits and Asymptotic Cones
77
ou L
88
MºPolyhedral Complexes
97
soonds LVO Jo Suſtuſ lulodf
98
Group Actions and QuasiIsometries
132
Further Properties of Hyperbolic Groups
460
Semihyperbolic Groups
471
Subgroups of Cocompact Groups of Isometries
481
Amalgamating Groups of Isometries
496
of Doubles
506
Complexes of Groups
520
Complexes of Groups
534
The Fundamental Group of a Complex of Groups
546

CATk Spaces
157
SL los popunog º Jo Ouluo
177
8
260
Fundamental Groups and Coverings
314
Aspects of the Geometry of Group Actions
397
NonPositive Curvature and Group Theory
438
Local Developments of a Complex of Groups
555
Coverings of Complexes of Groups
566
Groupoids of local Isometries
584
Etale Groupoids Homomorphisms and Equivalences 59 4
594
The Fundamental Group and Coverings of Étale Groupoids
604
Proof of the Main Theorem
616

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Page 630 - P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Ann.

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