Lectures on coarse geometry / John Roe.

Format:

Book

Author/Creator:

Roe, John, 1959-

Series:

University lecture series (Providence, R.I.) ; 31.

University lecture series, 1047-3998 ; v. 31

Language:

English

Subjects (All):

Metric spaces.

Algebraic topology.

Physical Description:

vii, 175 pages : illustrations ; 26 cm.

Other Title:

Coarse geometry

Place of Publication:

Providence, R.I. : American Mathematical Society, [2003]

Contents:

Chapter 1. Metric Spaces 1

1.1. Legendre on hyperbolic geometry 1

1.2. Metric spaces and length spaces 3

1.3. The coarse perspective on metric spaces 5

1.4. Groups and leaves 7

1.5. Trees and complexes 11

1.6. Hyperbolic space 13

1.7. Nilpotent examples

Heisenberg group 18

Chapter 2. Coarse Spaces 21

2.1. The abstract notion of coarse structure 21

2.2. Topological coarse structure 26

2.3. The Higson corona 29

2.4. Metrization of coarse structures 33

2.5. Hyperbolization 35

Chapter 3. Growth and amenability 39

3.1. Bounded geometry 39

3.3. Amenable spaces 44

3.4. Examples of amenable and nonamenable spaces 47

3.5. Amenable groups 51

3.6. Folner's Theorem 55

3.7. Amenability and analysis 58

Chapter 4. Translation Algebras 59

4.1. Translation Algebras 59

4.2. Amenability and the Translation Algebra 60

4.3. Finiteness of Group Algebras 63

4.4. Translation C*-Algebras 66

4.5. Translation Algebras as Crossed Products 68

Chapter 5. Coarse Algebraic Topology 71

5.1. Coarse cohomology theory 71

5.2. Product structure on coarse theory 76

5.3. Computation of coarse cohomology 78

5.4. Covers, nerves and metrization 80

5.5. Coarse homology theories 82

5.6. The Coarse Baum-Connes conjecture 84

Chapter 6. Coarse Negative Curvature 87

6.1. Curvature conditions 87

6.2. The Rips property and Gromov hyperbolicity 88

6.3. Controlling quasigeodesics 92

6.4. The Gromov boundary of a hyperbolic space 93

6.5. Bolicity 97

Chapter 7. Limits of Metric Spaces 99

7.1. Convergence of metric spaces 99

7.2. The rescaled limit of metric spaces 101

7.3. Groups of polynomial growth 104

7.4. Ultralimits 105

7.5. Asymptotic Cones 107

Chapter 8. Rigidity 111

8.1. What is rigidity? 111

8.2. The quasi-isometry group of hyperbolic space 112

8.3. Proof of Mostow Rigidity 117

8.4. Quasi-Isometric Rigidity For Products of Hyperbolic Spaces 121

Chapter 9. Asymptotic Dimension 129

9.1. The asymptotic dimension of a coarse space 129

9.2. Composition properties of asymptotic dimension 132

9.3. More Examples 136

9.4. Analytic implications of finite asymptotic dimension 139

Chapter 10. Groupoids and coarse geometry 141

10.1. Reminders about topological groupoids 141

10.2. The pair product and the Stone-Cech boundary 143

10.3. The translation groupoid of a coarse space 145

10.4. Translation groupoid and translation algebra 148

Chapter 11. Coarse Embeddability 151

11.1. Coarse embedding 151

11.2. Kernels and embeddings in Hilbert space 153

11.3. Embeddability and Property T 157

11.4. Property T and coarse equivalence 162

11.5. Property A and exactness for group C*-algebras 165.

Notes:

Includes bibliographical references (pages 173-175).

ISBN:

0821833324

OCLC:

52814432