ISBN: 3-540-64324-9
TITLE: Metric Spaces of Non-Positive Curvature
AUTHOR: Bridson, Martin R.; Haefliger, Andre
TOC:

Introduction 
Part I. Geodesic Metric Spaces 1 
1. Basic Concepts 2 
Metric Spaces 2 
Geodesics 4 
Angles 8 
The Length of a Curve 12 
2. The Model Spaces M^n_k 15 
Euclidean n-Space E^n 15 
The n-Sphere S^n 16 
Hyperbolic n-Space H^n 18 
The Model Spaces M^n_k 23 
Alexandrov's Lemma 24 
The Isometry Groups Isom(M^n_k) 26 
Approximate Midpoints 30 
3. Length Spaces 32 
Length Metrics 32 
The Hopf-Rinow Theorem 35 
Riemannian Manifolds as Metric Spaces 39 
Length Metrics on Covering Spaces 42 
Manifolds of Constant Curvature 45 
4. Normed Spaces 47 
Hilbert Spaces 47 
Isometries of Normed Spaces 51 
l^p Spaces 53 
5. Some Basic Constructions 56 
Products 56 
kappa-Cones 59 
Spherical Joins 63 
Quotient Metrics and Gluing 64 
Limits of Metric Spaces 70 
Ultralimits and Asymptotic Cones 77 
6. More on the Geometry of M^n_k 81 
The Klein Model for H^n 81 
The Mbius Group 84 
The Poincar Ball Model for H^n 86 
The Poincar Half-Space Model for H^n 90 
Isometries of H^2 91 
M^n_k as a Riemannian Manifold 92 
7. M_k-Polyhedral Complexes 97 
Metric Simplicial Complexes 97 
Geometric Links and Cone Neighbourhoods 102 
The Existence of Geodesics 105 
The Main Argument 108 
Cubical Complexes 111 
M_k-Polyhedral Complexes 112 
Barycentric Subdivision 115 
More on the Geometry of Geodesics 118 
Alternative Hypotheses 122 
Appendix: Metrizing Abstract Simplicial Complexes 123 
8. Group Actions and Quasi-Isometries 131 
Group Actions on Metric Spaces 131 
Presenting Groups of Homeomorphisms 134 
Quasi-Isometries 138 
Some Invariants of Quasi-Isometry 142 
The Ends of a Space 144 
Growth and Rigidity 148 
Quasi-Isometries of the Model Spaces 150 
Approximation by Metric Graphs 152 
Appendix: Combinatorial 2-Complexes 153 
Part II. CAT(kappa) Spaces 157 
1. Definitions and Characterizations of CAT(kappa) Spaces 158 
The CAT(kappa) Inequality 158 
Characterizations of CAT(kappa) Spaces 161 
CAT(kappa) Implies CAT(kappa') if kappa <= kappa' 165 
Simple Examples of CAT(kappa) Spaces 167 
Historical Remarks 168 
Appendix: The Curvature of Riemannian Manifolds 169 
2. Convexity and Its Consequences 175 
Convexity of the Metric 175 
Convex Subspaces and Projection 176 
The Centre of a Bounded Set 178 
Flat Subspaces 180 
3. Angles, Limits, Cones and Joins 184 
Angles in CAT(kappa) Spaces 184 
4-Point Limits of CAT(kappa) Spaces 186 
Cones and Spherical Joins 188 
The Space of Directions 190 
4. The Cartan-Hadamard Theorem 193 
Local-to-Global 193 
An Exponential Map 196 
Alexandrov's Patchwork 199 
Local Isometries and pi_1-Injectivity 200 
Injectivity Radius and Systole 202 
5. M_k-Polyhedral Complexes of Bounded Curvature 205 
Characterizations of Curvature <= kappa 206 
Extending Geodesics 207 
Flag Complexes 210 
Constructions with Cubical Complexes 212 
Two-Dimensional Complexes 215 
Subcomplexes and Subgroups in Dimension 2 216 
Knot and Link Groups 220 
From Group Presentations to Negatively Curved 2-Complexes 224 
6. Isometries of CAT(0) Spaces 228 
Individual Isometries 228 
On the General Structure of Groups of Isometries 233 
Clifford Translations and the Euclidean de Rham Factor 235 
The Group of Isometries of a Compact Metric Space 
of Non-Positive Curvature 237 
A Splitting Theorem 239 
7. The Flat Torus Theorem 244 
The Flat Torus Theorem 244 
Cocompact Actions and the Solvable Subgroup Theorem 247 
Proper Actions That Are Not Cocompact 250 
Actions That Are Not Proper 254 
Some Applications to Topology 254 
8. The Boundary at Infinity of a CAT(0) Space 260 
Asymptotic Rays and the Boundary partial X 260 
The Cone Topology on X = X U partial X 263 
Horofunctions and Busemann Functions 267 
Characterizations of Horofunctions 271 
Parabolic Isometries 274 
9. The Tits Metric and Visibility Spaces 277 
Angles in X 278 
The Angular Metric 279 
The Boundary (partial X, ?) is a CAT(1) Space 285 
The Tits Metric 289 
How the Tits Metric Determines Splittings 291 
Visibility Spaces 294 
10. Symmetric Spaces 299 
Real, Complex and Quaternionic Hyperbolic n-Spaces 300 
The Curvature of KH^n 304 
The Curvature of Distinguished Subspaces of KH^n 306 
The Group of Isometries of KH^n 307 
The Boundary at Infinity and Horospheres in KH^n 309 
Horocyclic Coordinates and Parabolic Subgroups for KH^n 311 
The Symmetric Space P(n,R) 314 
P(n,R) as a Riemannian Manifold 314 
The Exponential Map exp: M(n,R) -> GL(n,R) 316 
P(n,R) is a CAT(0) Space 318 
Flats, Regular Geodesics and Weyl Chambers 320 
The Iwasawa Decomposition of GL(n,R) 323 
The Irreducible Symmetric space P(n,R)_1 324 
Reductive Subgroups of GL(n,R) 327 
Semi-Simple Isometries 331 
Parabolic Subgroups and Horospherical Decompositions of P(n,R) 332 
The Tits Boundary of P(n,R)_1 is a Spherical Building 337 
partial_T P(n,R) in the Language of Flags and Frames 340 
Appendix: Spherical and Euclidean Buildings 342 
11. Gluing Constructions 347 
Gluing CAT(kappa) Spaces Along Convex Subspaces 347 
Gluing Using Local Isometries 350 
Equivariant Gluing 355 
Gluing Along Subspaces that are not Locally Convex 359 
Truncated Hyperbolic Spaces 362 
12. Simple Complexes of Groups 367 
Stratified Spaces 368 
Group Actions with a Strict Fundamental Domain 372 
Simple Complexes of Groups: Definition and Examples 375 
The Basic Construction 381 
Local Development and Curvature 387 
Constructions Using Coxeter Groups 391 
Part III. Aspects of the Geometry of Group Actions 397 
H. delta-Hyperbolic Spaces 398 
1. Hyperbolic Metric Spaces 399 
The Slim Triangles Condition 399 
Quasi-Geodesics in Hyperbolic Spaces 400 
k-Local Geodesics 405 
Reformulations of the Hyperbolicity Condition 407 
2. Area and Isoperimetric Inequalities 414 
A Coarse Notion of Area 414 
The Linear Isoperimetric Inequality and Hyperbolicity 417 
Sub-Quadratic Implies Linear 422 
More Refined Notions of Area 425 
3. The Gromov Boundary of a delta-Hyperbolic Space 427 
The Boundary partial X as a Set of Rays 427 
The Topology on X U partial X 429 
Metrizing partial X 432 
Non-Positive Curvature and Group Theory 438 
1. Isometries of CAT(0) Spaces 439 
A Summary of What We Already Know 439 
Decision Problems for Groups of Isometries 440 
The Word Problem 442 
The Conjugacy Problem 445 
2. Hyperbolic Groups and Their Algorithmic Properties 448 
Hyperbolic Groups 448 
Dehn's Algorithm 449 
The Conjugacy Problem 451 
Cone Types and Growth 455 
3. Further Properties of Hyperbolic Groups 459 
Finite Subgroups 459 
Quasiconvexity and Centralizers 460 
Translation Lengths 464 
Free Subgroups 467 
The Rips Complex 468 
4. Semihyperbolic Groups 471 
Definitions 471 
Basic Properties of Semihyperbolic Groups 473 
Subgroups of Semihyperbolic Groups 475 
5. Subgroups of Cocompact Groups of Isometries 481 
Finiteness Properties 481 
The Word, Conjugacy and Membership Problems 487 
Isomorphism Problems 491 
Distinguishing Among Non-Positively Curved 
Manifolds 494 
6. Amalgamating Groups of Isometries 496 
Amalgamated Free Products and HNN Extensions 497 
Amalgamating Along Abelian Subgroups 500 
Amalgamating Along Free Subgroups 503 
Subgroup Distortion and the Dehn Functions 
of Doubles 506 
7. Finite-Sheeted Coverings and Residual Finiteness 511 
Residual Finiteness 511 
Groups Without Finite Quotients 514 
C. Complexes of Groups 519 
1. Small Categories Without Loops (Scwols) 520 
Scwols and Their Geometric Realizations 521 
The Fundamental Group and Coverings 526 
Group Actions on Scwols 528 
The Local Structure of Scwols 531 
2. Complexes of Groups 534 
Basic Definitions 535 
Developability 538 
The Basic Construction 542 
3. The Fundamental Group of a Complex of Groups 546 
The Universal Group FG(Y) 546 
The Fundamental Group pi_1 (G(Y), sigma_0) 548 
A Presentation of pi_1 (G(Y), sigma_0) 549 
The Universal Covering of a Developable Complex of Groups 553 
4. Local Developments of a Complex of Groups 555 
The Local Structure of the Geometric Realization 555 
The Geometric Realization of the Local Development 557 
Local Development and Curvature 562 
The Local Development as a Scwol 564 
5. Coverings of Complexes of Groups 566 
Definitions 566 
The Fibres of a Covering 568 
The Monodromy 572 
A Appendix: Fundamental Groups and Coverings 
of Small Categories 573 
Basic Definitions 574 
The Fundamental Group 576 
Covering of a Category 579 
The Relationship with Coverings of Complexes of Groups 583 
G. Groupoids of local isometries 584 
1. Orbifolds 585 
Basic Definitions 585 
Coverings of Orbifolds 589 
Orbifolds with Geometric Structures 591 
2. tale Groupoids, Homomorphisms and Equivalences 594 
tale Groupoids 594 
Equivalences and Developability 597 
Groupoids of Local Isometries 601 
Statement of the Main Theorem 603 
3. The Fundamental Group and Coverings of tale Groupoids 604 
Equivalence and Homotopy of G-Paths 604 
The Fundamental Group pi_1 ((G,X), x_0) 607 
Coverings 609 
4. Proof of the Main Theorem 613 
Outline of the Proof 613 
G-Geodesics 614 
The Space X of G-Geodesics Issuing from 
a Base Point 616 
The Space X = X/G 617 
The Covering p : X -> X 618 
References 620 
Index 637 
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