Metric spaces of non-positive curvature / Martin R. Bridson, André Haefliger.
Series Grundlehren der mathematischen Wissenschaften, 319Editor: Berlin ; New York : Springer, c1999Descripción: xxi, 643 p. : ill. ; 24 cmISBN: 3540643249 (hardcover : alk. paper)Tema(s): Metric spaces | Geometry, DifferentialOtra clasificación: 53C23 (20F65 53C70 57M07) Recursos en línea: Publisher description | Table of contents onlyIntroduction VII -- 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 [15] -- Euclidean n-Space E" [15] -- The n-Sphere S" [16] -- Hyperbolic n-Space HI" [18] -- The Model Spaces M" [23] -- Alexandrov’s Lemma [24] -- The Isometry Groups Isom(M) [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] -- Spaces [53] -- 5. Some Basic Constructions [56] -- Products [56] -- k-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 [81] -- The Klein Model for H" [81] -- The Mobius Group [84] -- The Poincaré Ball Model for H" [86] -- The Poincaré Half-Space Model for H" [90] -- Isometnes of H [91] -- M" as a Riemannian Manifold [92] -- 7. Mk-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] -- Mk -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(k) Spaces [157] -- 1. Definitions and Characterizations of CAT(k) Spaces [158] -- The CAT(x) Inequality [158] -- Characterizations of CAT(x) Spaces [161] -- CAT(x) Implies CAT(x') if k < k' [165] -- Simple Examples of CAT(x) 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(/k) Spaces [184] -- 4-Point Limits of CAT(/k) 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't-Injectivity [200] -- Injectivity Radius and Systole [202] -- 5. Polyhedral Complexes of Bounded Curvature [205] -- Characterizations of Curvature < k [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(O) Space [260] -- Asymptotic Rays and the Boundary aX [260] -- The Cone Topology on X = X U aX [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 (aX, Z) 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 Quatemionic Hyperbolic n-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) —► 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)i [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)i is a Spherical Building [337] -- drP(n, R) in the Language of Flags and Frames [340] -- Appendix: Spherical and Euclidean Buildings -- 11. Gluing Constructions [347] -- Gluing CAT(k) 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. 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 S-Hyperbolic Space [427] -- The Boundary aX as a Set of Rays [427] -- The Topology on X U aX [429] -- Metrizing dX [432] -- T. Non-Positive Curvature and Group Theory [438] -- 1. Isometries of CAT(O) 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 48 [1] -- The Word, Conjugacy and Membership Problems [487] -- Isomorphism Problems [491] -- Distinguishing Among Non-Positively Curved [494] -- Manifolds -- 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 Finitenes [511] -- Residual Finiteness [511] -- Groups Without Finite Quotients [511] -- 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 7Ti(G(y) [546] -- A Presentation of 7Ti(G( J7) [548] -- The Universal Covering of a Developable Complex of Groups [549] -- 4. Local Developments of a Complex of Groups [553] -- 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 Smail 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] -- l.Orbifolds [585] -- Basic Definitions [585] -- Coverings of Orbifolds [589] -- Orbi folds with Geometric Structures [591] -- 2. Ftale Groupoids, Hosnomorphbtm and Equivalences [594] -- Etale Groupoids [594] -- Equivalences and Developability [597] -- Groupoids of Local Isometrics [601] -- Statement of the Main Theorem [603] -- 3. The Fundamental Group and Coverings of Étale Groopuid [604] -- Equivalence and Homotopy of G-Paths [604] -- The Fundamental Group [607] -- Coverings [609] -- 4. Proof of the Main Theorem [613] -- Outline of the Proof [613] -- Geodesics [614] -- The Space X of Geodeatcs Issuing from a Base Point [616] -- The Space X - X/G [617] -- The Covering [618] -- References [620] -- Index [637] --
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
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Instituto de Matemática, CONICET-UNS | Últimas adquisiciones | 53 B852 (Browse shelf) | Available | A-9391 |
Includes bibliographical references (p. 620-636) and index.
Introduction VII --
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 [15] --
Euclidean n-Space E" [15] --
The n-Sphere S" [16] --
Hyperbolic n-Space HI" [18] --
The Model Spaces M" [23] --
Alexandrov’s Lemma [24] --
The Isometry Groups Isom(M) [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] --
Spaces [53] --
5. Some Basic Constructions [56] --
Products [56] --
k-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 [81] --
The Klein Model for H" [81] --
The Mobius Group [84] --
The Poincaré Ball Model for H" [86] --
The Poincaré Half-Space Model for H" [90] --
Isometnes of H [91] --
M" as a Riemannian Manifold [92] --
7. Mk-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] --
Mk -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(k) Spaces [157] --
1. Definitions and Characterizations of CAT(k) Spaces [158] --
The CAT(x) Inequality [158] --
Characterizations of CAT(x) Spaces [161] --
CAT(x) Implies CAT(x') if k < k' [165] --
Simple Examples of CAT(x) 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(/k) Spaces [184] --
4-Point Limits of CAT(/k) 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't-Injectivity [200] --
Injectivity Radius and Systole [202] --
5. Polyhedral Complexes of Bounded Curvature [205] --
Characterizations of Curvature < k [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(O) Space [260] --
Asymptotic Rays and the Boundary aX [260] --
The Cone Topology on X = X U aX [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 (aX, Z) 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 Quatemionic Hyperbolic n-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) —► 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)i [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)i is a Spherical Building [337] --
drP(n, R) in the Language of Flags and Frames [340] --
Appendix: Spherical and Euclidean Buildings --
11. Gluing Constructions [347] --
Gluing CAT(k) 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. 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 S-Hyperbolic Space [427] --
The Boundary aX as a Set of Rays [427] --
The Topology on X U aX [429] --
Metrizing dX [432] --
T. Non-Positive Curvature and Group Theory [438] --
1. Isometries of CAT(O) 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 48 [1] --
The Word, Conjugacy and Membership Problems [487] --
Isomorphism Problems [491] --
Distinguishing Among Non-Positively Curved [494] --
Manifolds --
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 Finitenes [511] --
Residual Finiteness [511] --
Groups Without Finite Quotients [511] --
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 7Ti(G(y) [546] --
A Presentation of 7Ti(G( J7) [548] --
The Universal Covering of a Developable Complex of Groups [549] --
4. Local Developments of a Complex of Groups [553] --
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 Smail 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] --
l.Orbifolds [585] --
Basic Definitions [585] --
Coverings of Orbifolds [589] --
Orbi folds with Geometric Structures [591] --
2. Ftale Groupoids, Hosnomorphbtm and Equivalences [594] --
Etale Groupoids [594] --
Equivalences and Developability [597] --
Groupoids of Local Isometrics [601] --
Statement of the Main Theorem [603] --
3. The Fundamental Group and Coverings of Étale Groopuid [604] --
Equivalence and Homotopy of G-Paths [604] --
The Fundamental Group [607] --
Coverings [609] --
4. Proof of the Main Theorem [613] --
Outline of the Proof [613] --
Geodesics [614] --
The Space X of Geodeatcs Issuing from a Base Point [616] --
The Space X - X/G [617] --
The Covering [618] --
References [620] --
Index [637] --
MR, REVIEW #
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