Groups of circle diffeomorphisms / Andrés Navas.

Format:

Book

Author/Creator:

Navas, Andrés

Series:

Chicago lectures in mathematics

Chicago lectures in mathematics series

Language:

English

Subjects (All):

Diffeomorphisms.

Group actions (Mathematics).

Manifolds (Mathematics).

Physical Description:

xviii, 290 pages : illustrations ; 24 cm.

Place of Publication:

Chicago : University of Chicago Press, 2011.

Summary:

In recent years scholars from a variety of branches of mathematics have made several significant developments in the theory of group actions. Groups of Circle Diffeomorphisms systematically explores group actions on the simplest closed manifold, the circle. As the group of circle diffeomorphisms is an important subject in modern mathematics, this book will be of interest to those doing research in group theory, dynamical systems, low dimensional geometry and topology, and foliation theory. The book is mostly self-contained and also includes numerous complementary exercises, making it an excellent textbook for undergraduate and graduate students. Book jacket.

Contents:

Notation and General Definitions xvii

1 Examples of Group Actions on the Circle 1

1.1 The Group of Rotations 1

1.2 The Group of Translations and the Affine Group 2

1.3 The Group PSL(2, ) 4

1.3.1 PSL(2, ) as the Möbius group 4

1.3.2 PSL(2, ) and the Liouville geodesic current 7

1.3.3 PSL(2, ) and the convergence property 9

1.4 Actions of Lie Groups 11

1.5 Thompson's Groups 12

1.5.1 Thurston's piecewise projective realization 15

1.5.2 Ghys-Sergiescu's smooth realization 18

2 Dynamics of Groups of Homeomorphisms 23

2.1 Minimal Invariant Sets 23

2.1.1 The case of the circle 23

2.1.2 The case of the real line 29

2.2 Some Combinatorial Results 30

2.2.1 Poincaré's theory 30

2.2.2 Rotation numbers and invariant measures 37

2.2.3 Faithful actions on the line 39

2.2.4 Free actions and Hölder's theorem 43

2.2.5 Translation numbers and quasi-invariant measures 49

2.2.6 An application to amenable, orderable groups 58

2.3 Invariant Measures and Free Groups 63

2.3.1 A weak version of the Tits alternative 63

2.3.2 A probabilistic viewpoint 69

3 Dynamics of Groups of Diffeomorphisms 80

3.1 Denjoy's Theorem 80

3.2 Sacksteder's Theorem 90

3.2.1 The classical version in class 91

3.2.2 The version for pseudogroups 96

3.2.3 A sharp version via Lyapunov exponents 103

3.3 Dummy's First Theorem: On the Existence of Exceptional Minimal Sets 110

3.3.1 The statement of the result 110

3.3.2 An expanding first-return map 112

3.3.3 Proof of the theorem 116

3.4 Dummy's Second Theorem: On the Space of Semiexceptional Orbits 119

3.4.1 The statement of the result 119

3.4.2 A criterion for distinguishing two different ends 123

3.4.3 End of the proof 127

3.5 Two Open Problems 130

3.5.1 Minimal actions 130

3.5.2 Actions with an exceptional minimal set 141

3.6 On the Smoothness of the Conjugacy between Groups of Diffeomorphisms 146

3.6.1 Sternberg's linearization theorem and conjugacies 147

3.6.2 The case of bi-Lipschitz conjugacies 152

4 Structure and Rigidity via Dynamical Methods 158

4.1 Abelian Groups of Diffeomorphisms 158

4.1.1 Kopell's lemma 158

4.1.2 Classifying Abelian group actions in class 164

4.1.3 Szekeres's theorem 165

4.1.4 Denjoy counterexamples 171

4.1.5 On intermediate regularities 182

4.2 Nilpotent Groups of Diffeomorphisms 192

4.2.1 The Plante-Thurston Theorems 192

4.2.2 On growth of groups of diffeomorphisms 195

4.2.3 Nilpotence, growth, and intermediate regularity 202

4.3 Polycyclic Groups of Diffeomorphisms 209

4.4 Solvable Groups of Diffeomorphisms 211

4.4.1 Some examples and statements of results 211

4.4.2 The metabelian case 216

4.4.3 The case of the real line 220

4.5 On the Smooth Actions of Amenable Groups 223

5 Rigidity via Cohomological Methods 226

5.1 Thurston's Stability Theorem 226

5.2 Rigidity for Groups with Kazhdan's Property (T) 233

5.2.1 Kazhdan's property (T) 233

5.2.2 The statement of the result 240

5.2.3 Proof of the theorem 243

5.2.4 Relative property (T) and Haagerup's property 251

5.3 Superrigidity for Higher-Rank Lattice Actions 253

5.3.1 Statement of the result 253

5.3.2 Cohomological superrigidity 256

5.3.3 Superrigidity for actions on the circle 262

Appendix A Some Basic Concepts in Group Theory 267

Appendix B Invariant Measures and Amenable Groups 269

References 273.

Notes:

Includes bibliographical references (pages [273]-286) and index.

ISBN:

9780226569512

0226569519

OCLC:

646308529