Verfügbarkeit: Metric spaces of non-positive curvature

Metric spaces of non-positive curvature:

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Hauptverfasser:	Bridson, Martin R. 1964- (VerfasserIn), Haefliger, André 1929-2023 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin ; Heidelberg Springer [2010]
Schriftenreihe:	Grundlehren der mathematischen Wissenschaften 319
Schlagworte:	Nichtpositive Krümmung Metrischer Raum
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXI, 643 Seiten Diagramme
ISBN:	9783642083990

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Datensatz im Suchindex

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adam_text	Table of Contents Introduction .................................................. VII Parti. Geodesie Metric Spaces I 1. Basic Concepts ............................................ 2 Metric Spaces .............................................. 2 Geodesies ................................................. 4 Angles ................................................... 8 The Length of a Curve ....................................... 12 2. The Model Spaces M»K ...................................... 15 Euclidean л -Space E ........................................ 15 The п -Sphere S ............................................ 16 Hyperbolic n-Space H ...................................... 18 The Model Spaces MnK ....................................... 23 Alexandrov s Lemma ........................................ 24 The Isometry Groups Isom(Af£) ................................ 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 Hubert Spaces ............................................. 47 Isometries of Normed Spaces ................................. 51 V Spaces ................................................. 53 5. Some Basic Constructions ................................... 56 Products .................................................. 56 к -Cones .................................................. 59 XVI Table of Contents Spherical Joins ............................................. 63 Quotient Metrics and Gluing .................................. 64 Limits of Metric Spaces ...................................... 70 Ultralimits and Asymptotic Cones .............................. 77 6. More on the Geometry of MnK ................................ 81 The Klein Model for H ...................................... 81 The Möbius Group .......................................... 84 The Poincaré Ball Model for H ............................... 86 The Poincaré Half-Space Model forH .......................... 90 Isometries of H2 ............................................ 91 M as a Riemannian Manifold ................................. 92 7. ¿¿„-Polyhedral Complexes ................................... 97 Metric Simplicial Complexes .................................. 97 Geometric Links and Cone Neighbourhoods ...................... 102 The Existence of Geodesies ................................... 105 The Main Argument ......................................... 108 Cubical Complexes ......................................... Ill M^-Polyhedral Complexes .................................... 112 Barycentric Subdivision ...................................... 115 More on the Geometry of Geodesies ............................ 118 Alternative Hypotheses ...................................... 122 Appendix: Metrizing Abstract Simplicial Complexes ............... 123 8. Group Actions and Quasi-lsometries .......................... 131 Group Actions on Metric Spaces ............................... 131 Presenting Groups of Homeomorphisms ......................... 134 Quasi-lsometries ........................................... 138 Some Invariants of Quasi-Isometry ............................. 142 The Ends of a Space ......................................... 144 Growth and Rigidity ......................................... 148 Quasi-lsometries of the Model Spaces ........................... 150 Approximation by Metric Graphs .............................. 152 Appendix: Combinatorial 2-Complexes .......................... 153 Partii. САТ(к) Spaces 157 1. Definitions and Characterizations of САТ(к) Spaces ............. 158 The САЦ/с) Inequality ....................................... 158 Characterizations of САТ(ас) Spaces ............................ 161 CATOO Implies CATV) if к < к .............................. 165 Simple Examples of CAT^c) Spaces ............................ 167 Table of Contents XVII 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 САТ(к) Spaces ..................................... 184 4-Point Limits of CAT^c) 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 n i-Injectivity ............................. 200 Injectivity Radius and Systole ................................. 202 5. Мк -Polyhedral Complexes of Bounded Curvature ............... 205 Characterizations of Curvature < к ............................. 206 Extending Geodesies ........................................ 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 С AT(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 Rat 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 XVIII Table of Contents 8. The Boundary at Infinity of a CAT(O) Space .................... 260 Asymptotic Rays and the Boundary ЭХ .......................... 260 The Cone Topology on X = X U ЭХ ............................ 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 (ЭХ, Z) is a CAT(l) Space ........................ 285 The Tits Metric ............................................. 289 How the Tits Metric Determines Splittings ....................... 291 Visibility Spaces ............................................ 294 10. Symmetric Spaces ......................................... 299 Real, Complex and Quatemionic Hyperbolic η -Spaces .............. 300 The Curvature of KH ....................................... 304 The Curvature of Distinguished Subspaces of KH ................ 306 The Group of Isometries of KH ............................... 307 The Boundary at Infinity and Horospheres in KH ................. 309 Horocyclic Coordinates and Parabolic Subgroups for KH .......... 311 The Symmetric Space P(n, R) ................................. 314 P(n, R) as a Riemannian Manifold ............................. 314 The Exponential Map exp: M(n, R) -» GUn, R) .................. 316 P(n, R) is a CAIXO) Space .................................... 318 Flats, Regular Geodesies and Weyl Chambers ..................... 320 The Iwasawa Decomposition of GUn, R) ........................ 323 The Irreducible Symmetric space P(n, R)\| ....................... 324 Reductive Subgroups of GlÁn, R) .............................. 327 Semi-Simple Isometries ...................................... 331 Parabolic Subgroups and Horospherical Decompositions of P(n, R) ... 332 The Tits Boundary of P{n, R)i is a Spherical Building .............. 337 9rP(n,R) in the Language of Flags and Frames ................... 340 Appendix: Spherical and Euclidean Buildings ..................... 342 11. Gluing Constructions ....................................... 347 Gluing САГХк) 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 Table of Contents XIX 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 Partili. Aspects of the Geometry of Group Actions 397 H. ¿-Hyperbolic Spaces ....................................... 398 1. Hyperbolic Metric Spaces ................................. 399 The Slim Triangles Condition ............................... 399 Quasi-Geodesics in Hyperbolic Spaces ........................ 400 Jfc-Local Geodesies ........................................ 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 ¿-Hyperbolic Space .............. 427 The Boundary ЭХ as a Set of Rays ........................... 427 The Topology on X U дХ ................................... 429 Metrizing дХ ............................................ 432 Г. Non-Positive Curvature and Group Theory .................... 438 1. Isometries of CATiO) 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 XX Table of Contents 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 FGQ?) ................................ 546 The Fundamental Group ni(G{y)y σ0) ........................ 548 A Presentation of п^СКУ), σ0) .............................. 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 Table of Contents XXI The Fibres of a Covering ................................... 568 The Monodramy ......................................... 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 Q. Groupoids of local Isometries ................................ 584 1. Orbifolds ............................................... 585 Basic Definitions ......................................... 585 Coverings of Orbifolds ..................................... 589 Orbifolds with Geometric Structures .......................... 591 2. Étale Groupoids, Homomorphisms and Equivalences .......... 594 Étale Groupoids .......................................... 594 Equivalences and Developability ............................. 597 Groupoids of Local Isometries ............................... 601 Statement of the Main Theorem .............................. 603 3. The Fundamental Group and Coverings of Étale Groupoids ----- 604 Equivalence and Homotopy of (/-Paths ........................ 604 The Fundamental Group nx((G, X),xo) ........................ 607 Coverings ............................................... 609 4. Proof of the Main Theorem ................................ 613 Outline of the Proof ....................................... 613 Ç-Geodesics ............................................. 614 The Space X of ¿/-Geodesies Issuing from a Base Point ........... 616 TheSpaceX = X/Q ....................................... 617 The Coveringp : X -» X ................................... 618 References ................................................... 620 Indes ........................................................ 637
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author	Bridson, Martin R. 1964- Haefliger, André 1929-2023
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spelling	Bridson, Martin R. 1964- Verfasser (DE-588)121157423 aut Metric spaces of non-positive curvature Martin R. Bridson ; André Haefliger Berlin ; Heidelberg Springer [2010] © 2010 XXI, 643 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 319 Nichtpositive Krümmung (DE-588)4128763-0 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 s Nichtpositive Krümmung (DE-588)4128763-0 s DE-604 Haefliger, André 1929-2023 Verfasser (DE-588)121157377 aut Erscheint auch als Online-Ausgabe 978-3-662-12494-9 Grundlehren der mathematischen Wissenschaften 319 (DE-604)BV000000395 319 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020867895&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Bridson, Martin R. 1964- Haefliger, André 1929-2023 Metric spaces of non-positive curvature Grundlehren der mathematischen Wissenschaften Nichtpositive Krümmung (DE-588)4128763-0 gnd Metrischer Raum (DE-588)4169745-5 gnd
subject_GND	(DE-588)4128763-0 (DE-588)4169745-5
title	Metric spaces of non-positive curvature
title_auth	Metric spaces of non-positive curvature
title_exact_search	Metric spaces of non-positive curvature
title_full	Metric spaces of non-positive curvature Martin R. Bridson ; André Haefliger
title_fullStr	Metric spaces of non-positive curvature Martin R. Bridson ; André Haefliger
title_full_unstemmed	Metric spaces of non-positive curvature Martin R. Bridson ; André Haefliger
title_short	Metric spaces of non-positive curvature
title_sort	metric spaces of non positive curvature
topic	Nichtpositive Krümmung (DE-588)4128763-0 gnd Metrischer Raum (DE-588)4169745-5 gnd
topic_facet	Nichtpositive Krümmung Metrischer Raum
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020867895&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000395
work_keys_str_mv	AT bridsonmartinr metricspacesofnonpositivecurvature AT haefligerandre metricspacesofnonpositivecurvature

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