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Groups of Circle Diffeomorphisms : Chicago Lectures in Mathematics Series CLM - Andres Navas

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Groups of Circle Diffeomorphisms

By: Andres Navas

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Hardcover | 30 June 2011 | Edition Number 1

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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.

Industry Reviews

"This is a wonderful book about 'mildly' smooth actions of groups on the most important manifolds in mathematics: the circle and the line. Andres Navas draws upon the classical contributions of Poincare, Denjoy, Hoelder, 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."

--Rostislav Grigorchuk, Texas A&M University, and Etienne Ghys, Ecole Normale Superieure de Lyon

"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."

--Takashi Tsuboi, University of Tokyo

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Introduction	p. xi
Acknowledgments	p. xv
Notation and General Definitions	p. xvii
Examples of Group Actions on the Circle	p. 1
The Group of Rotations	p. 1
The Group of Translations and the Affine Group	p. 2
The Group PSL(2, )	p. 4
PSL(2, ) as the Möbius group	p. 4
PSL(2, ) and the Liouville geodesic current	p. 7
PSL(2, ) and the convergence property	p. 9
Actions of Lie Groups	p. 11
Thompson's Groups	p. 12
Thurston's piecewise projective realization	p. 15
Ghys-Sergiescu's smooth realization	p. 18
Dynamics of Groups of Homeomorphisms	p. 23
Minimal Invariant Sets	p. 23
The case of the circle	p. 23
The case of the real line	p. 29
Some Combinatorial Results	p. 30
Poincaré's theory	p. 30
Rotation numbers and invariant measures	p. 37
Faithful actions on the line	p. 39
Free actions and Hölder's theorem	p. 43
Translation numbers and quasi-invariant measures	p. 49
An application to amenable, orderable groups	p. 58
Invariant Measures and Free Groups	p. 63
A weak version of the Tits alternative	p. 63
A probabilistic viewpoint	p. 69
Dynamics of Groups of Diffeomorphisms	p. 80
Denjoy's Theorem	p. 80
Sacksteder's Theorem	p. 90
The classical version in class	p. 91
The version for pseudogroups	p. 96
A sharp version via Lyapunov exponents	p. 103
Dummy's First Theorem: On the Existence of Exceptional Minimal Sets	p. 110
The statement of the result	p. 110
An expanding first-return map	p. 112
Proof of the theorem	p. 116
Dummy's Second Theorem: On the Space of Semiexceptional Orbits	p. 119
The statement of the result	p. 119
A criterion for distinguishing two different ends	p. 123
End of the proof	p. 127
Two Open Problems	p. 130
Minimal actions	p. 130
Actions with an exceptional minimal set	p. 141
On the Smoothness of the Conjugacy between Groups of Diffeomorphisms	p. 146
Sternberg's linearization theorem and conjugacies	p. 147
The case of bi-Lipschitz conjugacies	p. 152
Structure and Rigidity via Dynamical Methods	p. 158
Abelian Groups of Diffeomorphisms	p. 158
Kopell's lemma	p. 158
Classifying Abelian group actions in class	p. 164
Szekeres's theorem	p. 165
Denjoy counterexamples	p. 171
On intermediate regularities	p. 182
Nilpotent Groups of Diffeomorphisms	p. 192
The Plante-Thurston Theorems	p. 192
On growth of groups of diffeomorphisms	p. 195
Nilpotence, growth, and intermediate regularity	p. 202
Polycyclic Groups of Diffeomorphisms	p. 209
Solvable Groups of Diffeomorphisms	p. 211
Some examples and statements of results	p. 211
The metabelian case	p. 216
The case of the real line	p. 220
On the Smooth Actions of Amenable Groups	p. 223
Rigidity via Cohomological Methods	p. 226
Thurston's Stability Theorem	p. 226
Rigidity for Groups with Kazhdan's Property (T)	p. 233
Kazhdan's property (T)	p. 233
The statement of the result	p. 240
Proof of the theorem	p. 243
Relative property (T) and Haagerup's property	p. 251
Superrigidity for Higher-Rank Lattice Actions	p. 253
Statement of the result	p. 253
Cohomological superrigidity	p. 256
Superrigidity for actions on the circle	p. 262
Some Basic Concepts in Group Theory	p. 267
Invariant Measures and Amenable Groups	p. 269
References	p. 273
Index	p. 287
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Groups of Circle Diffeomorphisms

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