Metric spaces of non-positive curvature
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|---|---|---|---|
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| 005 | 20220705 | ||
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| 008 | 110112s2010 gw |||| |||| 00||| eng d | ||
| 020 | |a 9783642083990 |c softcover |9 978-3-642-08399-0 | ||
| 035 | |a (OCoLC)706965222 | ||
| 035 | |a (DE-599)BVBBV036952876 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
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| 084 | |a MAT 546f |2 stub | ||
| 100 | 1 | |a Bridson, Martin R. |d 1964- |e Verfasser |0 (DE-588)121157423 |4 aut | |
| 245 | 1 | 0 | |a Metric spaces of non-positive curvature |c Martin R. Bridson ; André Haefliger |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2010] | |
| 264 | 4 | |c © 2010 | |
| 300 | |a XXI, 643 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 319 | |
| 650 | 0 | 7 | |a Nichtpositive Krümmung |0 (DE-588)4128763-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Metrischer Raum |0 (DE-588)4169745-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Metrischer Raum |0 (DE-588)4169745-5 |D s |
| 689 | 0 | 1 | |a Nichtpositive Krümmung |0 (DE-588)4128763-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Haefliger, André |d 1929-2023 |e Verfasser |0 (DE-588)121157377 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-662-12494-9 |
| 830 | 0 | |a Grundlehren der mathematischen Wissenschaften |v 319 |w (DE-604)BV000000395 |9 319 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-020867895 | |
Datensatz im Suchindex
| _version_ | 1819764858268155904 |
|---|---|
| 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
|
| any_adam_object | 1 |
| author | Bridson, Martin R. 1964- Haefliger, André 1929-2023 |
| author_GND | (DE-588)121157423 (DE-588)121157377 |
| author_facet | Bridson, Martin R. 1964- Haefliger, André 1929-2023 |
| author_role | aut aut |
| author_sort | Bridson, Martin R. 1964- |
| author_variant | m r b mr mrb a h ah |
| building | Verbundindex |
| bvnumber | BV036952876 |
| classification_rvk | SK 280 SK 370 |
| classification_tum | MAT 546f |
| ctrlnum | (OCoLC)706965222 (DE-599)BVBBV036952876 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036952876 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-24T00:17:38Z |
| institution | BVB |
| isbn | 9783642083990 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020867895 |
| oclc_num | 706965222 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-20 |
| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-20 |
| physical | XXI, 643 Seiten Diagramme |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| 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 |