Introduction

Acknowledgments

Notation and General Definitions

1 Examples of Group Actions on the Circle
1.1 The Group of Rotations

1.2 The Group of Translations and the Affine Group

1.3 The Group PSL(2, R)

1.3.1 PSL(2, R) as the Möbius group

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

1.3.3 PSL(2, R) and the convergence property

1.4 Actions of Lie Groups

1.5 Thompson’s Groups

1.5.1 Thurston’s piecewise projective realization

1.5.2 Ghys-Sergiescu’s smooth realization

2 Dynamics of Groups of Homeomorphisms
2.1 Minimal Invariant Sets

2.1.1 The case of the circle

2.1.2 The case of the real line

2.2 Some Combinatorial Results

2.2.1 Poincaré’s theory

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 Invariant Measures and Free Groups

2.3.1 A weak version of the Tits alternative

2.3.2 A probabilistic viewpoint

3 Dynamics of Groups of Diffeomorphisms
3.1 Denjoy’s Theorem

3.2 Sacksteder’s Theorem

3.2.1 The classical version in class C1+Lip

3.2.2 The C1 version for pseudogroups

3.2.3 A sharp C1 version via Lyapunov exponents

3.3 Duminy’s First Theorem: On the Existence of Exceptional Minimal Sets

3.3.1 The statement of the result

3.3.2 An expanding first-return map

3.3.3 Proof of the theorem

3.4 Duminy’s Second Theorem: On the Space of Semiexceptional Orbits

3.4.1 The statement of the result

3.4.2 A criterion for distinguishing two different ends

3.4.3 End of the proof

3.5 Two Open Problems

3.5.1 Minimal actions

3.5.2 Actions with an exceptional minimal set

3.6 On the Smoothness of the Conjugacy between Groups of Diffeomorphisms

3.6.1 Sternberg’s linearization theorem and C1 conjugacies

3.6.2 The case of bi-Lipschitz conjugacies

4 Structure and Rigidity via Dynamical Methods
4.1 Abelian Groups of Diffeomorphisms

4.1.1 Kopell’s lemma

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 Nilpotent Groups of Diffeomorphisms

4.2.1 The Plante-Thurston Theorems

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 Solvable Groups of Diffeomorphisms

4.4.1 Some examples and statements of results

4.4.2 The metabelian case

4.4.3 The case of the real line

4.5 On the Smooth Actions of Amenable Groups

5 Rigidity via Cohomological Methods
5.1 Thurston’s Stability Theorem

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

5.2.1 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 Superrigidity for Higher-Rank Lattice Actions

5.3.1 Statement of the result

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