# Groups of Circle Diffeomorphisms

- Contents
- Review Quotes

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

**Rostislav Grigorchuk, Texas A&M University, and Etienne Ghys, École Normale Supérieure de Lyon**

“This is a wonderful book about ‘mildly’ smooth actions of groups on the most important manifolds in mathematics: the circle and the line. Andrés Navas draws upon the classical contributions of Poincaré, Denjoy, Hölder, Plante, Thompson, Sacksteder, and Duminy, as well as the relatively recent achievements of Margulis and Witte Morris, to offer the first book-length exploration of this topic. The analytic techniques, the dynamical point of view, and the algebraic nature of objects considered here produce a blend of beautiful mathematics that will be used by researchers in several areas of science.”

**Takashi Tsuboi, University of Tokyo**

“*Groups of Circle Diffeomorphisms* provides a great overview of the research on differentiable group actions on the circle. Navas’s book will appeal to those doing research on differential topology, transformation groups, dynamical systems, foliation theory, and representation theory, and will be a solid base for those who want to further attack problems of group actions on higher dimensional manifolds or of geometric group theory.”

**Physical Sciences: **
Theoretical Physics

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