# Groups of Circle Diffeomorphisms

# Groups of Circle Diffeomorphisms

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.

312 pages | 24 line drawings | 6 x 9 | © 2011

Chicago Lectures in Mathematics

Physical Sciences: Theoretical Physics

## Reviews

## Table of Contents

Acknowledgments

Notation and General Definitions

1 Examples of Group Actions on the Circle

1.2 The Group of Translations and the Affine Group

1.3 The Group PSL(2, R)

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

1.3.3 PSL(2, R) and the convergence property

1.5 Thompson’s Groups

1.5.2 Ghys-Sergiescu’s smooth realization

2 Dynamics of Groups of Homeomorphisms

2.1.2 The case of the real line

2.2.2 Rotation numbers and invariant measures

2.2.3 Faithful actions on the line

2.2.4 Free actions and Hölder’s theorem

2.2.5 Translation numbers and quasi-invariant measures

2.2.6 An application to amenable, orderable groups

2.3.2 A probabilistic viewpoint

3 Dynamics of Groups of Diffeomorphisms

3.2 Sacksteder’s Theorem

3.2.2 The C1 version for pseudogroups

3.2.3 A sharp C1 version via Lyapunov exponents

3.3.2 An expanding first-return map

3.3.3 Proof of the theorem

3.4.2 A criterion for distinguishing two different ends

3.4.3 End of the proof

3.5.2 Actions with an exceptional minimal set

3.6.2 The case of bi-Lipschitz conjugacies

4 Structure and Rigidity via Dynamical Methods

4.1.2 Classifying Abelian group actions in class C2

4.1.3 Szekeres’s theorem

4.1.4 Denjoy counterexamples

4.1.5 On intermediate regularities

4.2.2 On growth of groups of diffeomorphisms

4.2.3 Nilpotence, growth, and intermediate regularity

4.3 Polycyclic Groups of Diffeomorphisms

4.4.2 The metabelian case

4.4.3 The case of the real line

5 Rigidity via Cohomological Methods

5.2 Rigidity for Groups with Kazhdan’s Property (T)

5.2.2 The statement of the result

5.2.3 Proof of the theorem

5.2.4 Relative property (T) and Haagerup’s property

5.3.2 Cohomological superrigidity

5.3.3 Superrigidity for actions on the circle

Appendix A Some Basic Concepts in Group Theory

Appendix B Invariant Measures and Amenable Groups

References

Index

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