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Groups of Circle Diffeomorphisms

Andrés Navas

Groups of Circle Diffeomorphisms

Andrés Navas

312 pages | 24 line drawings | 6 x 9 | © 2011
Cloth $52.00 ISBN: 9780226569512 Published June 2011
E-book $10.00 to $52.00 About E-books ISBN: 9780226569505 Published June 2011

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.



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


Review Quotes
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.”

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