# Groups acting on the circle

## Summary

The objective of the course is to study actions of the circle by homeomorphisms and by diffeomorphisms. In part, this will be a pretext to study some fundamental notions related to group theory and to dynamical systems.

1. Preliminaries on group actions and topological dynamics. Minimality. Existence of invariant measures for continuous maps on compact spaces.
2. Circle homeomorphisms. Rotation number theory and Denjoy's theorem
3. Introduction to amenability of groups Definition with invariant probability measures on compact spaces. Stability under group-theoretic operations. Free groups are not amenable. Notion of ping pong pair of homeomorphisms of a space and the ping-pong lemma. Amenability for topological groups and Haar measure on compact groups.
4. General theory of groups of circle homeomorphisms . Basic trichotomy: finite orbit, minimal actions, and exceptional minimal set. Notion of semiconjugacy. Margulis' theorem on existence of free subgroups. Derivation of the Tits' alternative for SL(2, R).
5. Left-orderable groups and actions on the real line. Definitions and basic properties. Holder's theorem. Locally indicable groups are left-orderable. The space of left orders and application to amenable groups (Witte-Morris' theorem).
6. Elements of differentiable theory. Thurston's stability theorem. Koppel's lemma. Smooth actions of nilpotent groups (Plante-Thurston theorem).
7. Actions of lattices. Witte-Morris theorem's for actions of SL(3, Z). Statement of Zimmer's conjecture and history of results (statements only).

## Exercises

A list of exercise is available here . It will be updated from time to time (last update: 13/06/2019). Handing in solutions is not required, but if you are willing to do so I'll be happy to give feedback. Questions on the exercises are also welcome (by email or in person). Solving various exercises from each section is the best possile preparation for the exam.

## References

•  E. Ghys. Groups acting on the circle. Enseign. Math. (2) 47 (2001), no. 3-4, 329-407. (a link to a pdf version can be found here ) Main reference for Chapter 2 and 4.
•  A. Navas Groups of circle diffeomorphisms. Chicago lectures in mathematics. ( Main reference for Denjoy's theorem and for Chapter 6. It is also a good reading for Chapters 2 and 4, and a good source of exercises).
•  A. Clay, D. Rolfsen, Ordered groups and topology AMS Graduate studies in Mathematics, vol. 176. (Reference for Chapter 5)
•  B. Deroin, A. Navas, C. Rivas. Groups, order and dynamics. ( For chapter 5). A preprint version is freely available here .