Elements of asymptotic geometry:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Society
2007
|
| Series: | EMS monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XII, 200 S. graph. Darst. |
| ISBN: | 9783037190364 3037190361 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text |
Contents
Preface
xi
1
Hyperbolic geodesic spaces
1
1.1
Geodesic metric spaces
. 1
1.2
Hyperbolic geodesic spaces
. 2
1.3
Stability of geodesies
. 4
1.4
Supplementary results and remarks
. 6
2
The boundary at infinity
9
2.1
5-inequality and hyperbolic spaces
. 9
2.2
The boundary at infinity of hyperbolic spaces
. 12
2.3
Local self-similarity of the boundary
. 16
2.4
Supplementary results and remarks
. 19
3
Busemann functions on hyperbolic spaces
23
3.1
Busemann functions
. 23
3.2
Gromov products based at infinity
. 26
3.3
Visual metrics based at infinity
. 29
3.4
Supplementary results and remarks
. 30
4
Morphisms of hyperbolic spaces
35
4.1
Morphisms of metric spaces and hyperbolicity
. 36
4.2
Cross-difference triples and cross-differences
. 39
4.3
PQ-isometric maps
. 41
4.4
Quasi-isometric maps of hyperbolic geodesic spaces
. 42
4.5
Supplementary results and remarks
. 44
5 Quasi-Möbius
and quasi-symmetric maps
49
5.1
Cross-ratios
. 49
5.2 Quasi-Möbius
and quasi-symmetric maps
. 50
5.3
Supplementary results and remarks
. 56
5.4
Summary
. 63
viii Contents
6
Hyperbolic approximation of metric spaces
69
6.1
Construction
. 69
6.2
Geodesies in a hyperbolic approximation
. 70
6.3
The boundary at infinity of a hyperbolic approximation
. 74
6.4
Supplementary results and remarks
. 77
7
Extension theorems
81
7.1
Extension theorem for bilipschitz maps
. 81
7.2
Extension theorem for quasi-symmetric maps
. 84
7.3
Extension theorem for
quasi-Möbius
maps
. 87
7.4
Supplementary results and remarks
. 95
8
Embedding theorems
97
8.1
Assouad embedding theorem
. 97
8.2
Bonk-Schramm embedding theorem
. 100
8.3
Supplementary results and remarks
. 102
9
Basics of dimension theory
107
9.1
Various dimensions
. 107
9.2
Constructions
.
Ill
9.3
P-dimensions
. 117
9.4
The
monotonicity
theorem
. 121
9.5
The product theorem
. 122
9.6
The saturation of families
. 122
9.7
The finite union theorem
. 123
9.8
Sperner lemma
. 124
9.9
Supplementary results and remarks
. 126
10
Asymptotic dimension
129
10.1
Estimates from below
. 129
10.2
Estimates from above
. 130
10.3
Embedding of H2 into a product of two trees
. 132
10.4
Supplementary results and remarks
. 134
11
Linearly controlled metric dimension: Basic properties
137
11.1
Separated sequences of colored coverings
. 138
11.2
Quasi-symmetry
invariance
of ^-dim
. 141
11.3
Supplementary results and remarks
. 145
12
Linearly controlled metric dimension: Applications
147
12.1
Embedding into the product of trees
. 147
12.2
¿-dimension of locally self-similar spaces
. 154
12.3
Applications to hyperbolic spaces
. 156
Contents ix
12.4
Supplementary results and remarks
. 157
13
Hyperbolic dimension
159
13.1
Large scale doubling sets
. 159
13.2
Definition of the hyperbolic dimension
. 160
13.3
Hyperbolic dimension of hyperbolic spaces
. 162
13.4
Applications to nonembedding results
. 164
13.5
Supplementary results and remarks
. 166
14
Hyperbolic rank and subexponential corank
167
14.1
Hyperbolic rank
. 167
14.2
Subexponential corank
. 169
14.3
Applications to nonembedding results
. 175
14.4
Subexponential corank versus hyperbolic dimension
. 175
14.5
Supplementary results and remarks
. 177
Appendix. Models of the hyperbolic space H"
181
A.
1
The pseudo-spherical model
. 181
A.2 The unit disc model
. 182
A.3 The upper half-plane model
. 183
A.4 The solvable group model
. 185
A.5 Generalizations to an arbitrary dimension
. 186
A.6
Möbius
transformations
. 187
A.7 Cross-ratio
. 188
Bibliography
193
Index
197 |
| adam_txt |
Contents
Preface
xi
1
Hyperbolic geodesic spaces
1
1.1
Geodesic metric spaces
. 1
1.2
Hyperbolic geodesic spaces
. 2
1.3
Stability of geodesies
. 4
1.4
Supplementary results and remarks
. 6
2
The boundary at infinity
9
2.1
5-inequality and hyperbolic spaces
. 9
2.2
The boundary at infinity of hyperbolic spaces
. 12
2.3
Local self-similarity of the boundary
. 16
2.4
Supplementary results and remarks
. 19
3
Busemann functions on hyperbolic spaces
23
3.1
Busemann functions
. 23
3.2
Gromov products based at infinity
. 26
3.3
Visual metrics based at infinity
. 29
3.4
Supplementary results and remarks
. 30
4
Morphisms of hyperbolic spaces
35
4.1
Morphisms of metric spaces and hyperbolicity
. 36
4.2
Cross-difference triples and cross-differences
. 39
4.3
PQ-isometric maps
. 41
4.4
Quasi-isometric maps of hyperbolic geodesic spaces
. 42
4.5
Supplementary results and remarks
. 44
5 Quasi-Möbius
and quasi-symmetric maps
49
5.1
Cross-ratios
. 49
5.2 Quasi-Möbius
and quasi-symmetric maps
. 50
5.3
Supplementary results and remarks
. 56
5.4
Summary
. 63
viii Contents
6
Hyperbolic approximation of metric spaces
69
6.1
Construction
. 69
6.2
Geodesies in a hyperbolic approximation
. 70
6.3
The boundary at infinity of a hyperbolic approximation
. 74
6.4
Supplementary results and remarks
. 77
7
Extension theorems
81
7.1
Extension theorem for bilipschitz maps
. 81
7.2
Extension theorem for quasi-symmetric maps
. 84
7.3
Extension theorem for
quasi-Möbius
maps
. 87
7.4
Supplementary results and remarks
. 95
8
Embedding theorems
97
8.1
Assouad embedding theorem
. 97
8.2
Bonk-Schramm embedding theorem
. 100
8.3
Supplementary results and remarks
. 102
9
Basics of dimension theory
107
9.1
Various dimensions
. 107
9.2
Constructions
.
Ill
9.3
P-dimensions
. 117
9.4
The
monotonicity
theorem
. 121
9.5
The product theorem
. 122
9.6
The saturation of families
. 122
9.7
The finite union theorem
. 123
9.8
Sperner lemma
. 124
9.9
Supplementary results and remarks
. 126
10
Asymptotic dimension
129
10.1
Estimates from below
. 129
10.2
Estimates from above
. 130
10.3
Embedding of H2 into a product of two trees
. 132
10.4
Supplementary results and remarks
. 134
11
Linearly controlled metric dimension: Basic properties
137
11.1
Separated sequences of colored coverings
. 138
11.2
Quasi-symmetry
invariance
of ^-dim
. 141
11.3
Supplementary results and remarks
. 145
12
Linearly controlled metric dimension: Applications
147
12.1
Embedding into the product of trees
. 147
12.2
¿-dimension of locally self-similar spaces
. 154
12.3
Applications to hyperbolic spaces
. 156
Contents ix
12.4
Supplementary results and remarks
. 157
13
Hyperbolic dimension
159
13.1
Large scale doubling sets
. 159
13.2
Definition of the hyperbolic dimension
. 160
13.3
Hyperbolic dimension of hyperbolic spaces
. 162
13.4
Applications to nonembedding results
. 164
13.5
Supplementary results and remarks
. 166
14
Hyperbolic rank and subexponential corank
167
14.1
Hyperbolic rank
. 167
14.2
Subexponential corank
. 169
14.3
Applications to nonembedding results
. 175
14.4
Subexponential corank versus hyperbolic dimension
. 175
14.5
Supplementary results and remarks
. 177
Appendix. Models of the hyperbolic space H"
181
A.
1
The pseudo-spherical model
. 181
A.2 The unit disc model
. 182
A.3 The upper half-plane model
. 183
A.4 The solvable group model
. 185
A.5 Generalizations to an arbitrary dimension
. 186
A.6
Möbius
transformations
. 187
A.7 Cross-ratio
. 188
Bibliography
193
Index
197 |
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| index_date | 2024-07-02T17:53:52Z |
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| isbn | 9783037190364 3037190361 |
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| spelling | Buyalo, Serguei Verfasser aut Elements of asymptotic geometry Sergei Buyalo ; Viktor Schroeder Zürich European Mathematical Society 2007 XII, 200 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier EMS monographs in mathematics Espaces hyperboliques Géométrie hyperbolique Geometry, Hyperbolic Hyperbolic spaces Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Globale Differentialgeometrie (DE-588)4021286-5 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 s Globale Differentialgeometrie (DE-588)4021286-5 s DE-604 Schroeder, Viktor Verfasser aut Erscheint auch als Buyalo, Serguei Elements of asymptotic geometry Online-Ausgabe 978-3-03719-536-9 (DE-604)BV036705926 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2968985&prov=M&dok_var=1&dok_ext=htm Inhaltstext text/html http://www.ems-ph.org/book.php?proj_nr=58&searchterm= Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015704292&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Buyalo, Serguei Schroeder, Viktor Elements of asymptotic geometry Espaces hyperboliques Géométrie hyperbolique Geometry, Hyperbolic Hyperbolic spaces Metrischer Raum (DE-588)4169745-5 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd |
| subject_GND | (DE-588)4169745-5 (DE-588)4021286-5 |
| title | Elements of asymptotic geometry |
| title_auth | Elements of asymptotic geometry |
| title_exact_search | Elements of asymptotic geometry |
| title_exact_search_txtP | Elements of asymptotic geometry |
| title_full | Elements of asymptotic geometry Sergei Buyalo ; Viktor Schroeder |
| title_fullStr | Elements of asymptotic geometry Sergei Buyalo ; Viktor Schroeder |
| title_full_unstemmed | Elements of asymptotic geometry Sergei Buyalo ; Viktor Schroeder |
| title_short | Elements of asymptotic geometry |
| title_sort | elements of asymptotic geometry |
| topic | Espaces hyperboliques Géométrie hyperbolique Geometry, Hyperbolic Hyperbolic spaces Metrischer Raum (DE-588)4169745-5 gnd Globale Differentialgeometrie (DE-588)4021286-5 gnd |
| topic_facet | Espaces hyperboliques Géométrie hyperbolique Geometry, Hyperbolic Hyperbolic spaces Metrischer Raum Globale Differentialgeometrie |
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