Groups and Dynamics II 21 de Abril 2014

Title: Rigidity of diffeomorphism groups.

Abstract: Let Diff(M) be the group of diffeomorphisms isotopic to the identity of a closed manifold M. As a discrete group Diff(M) is somewhat rigid: Diff(M) is a simple group and Filipkiewicz proved in 1982  that if Diff(M) and Diff(N) are isomorphic as abstract groups, then M and N should be diffeomorphic. 
I´ll talk about these theorems, about the concept of distortion in geometric group theory and about how to use this concept to prove that any homomorphism of groups P : Diff(M) —> DIff(N) is continuous.
This talk is going to be elementary and require only basic concepts of groups and manifolds. 

Title: Surface groups, representation spaces, and rigidity.

Abstract: Let G be the fundamental group of a closed surface S. In this talk, we discuss the space Hom(G, Homeo+(S^1)) of actions of G on the circle, equivalently the space of flat circle bundles over S. The Milnor-Wood inequality gives a lower bound on the number of connected components of this space (4g-3), but until very recently it was not known whether this bound was sharp. In fact, we still don´t know whether Hom(G, Homeo+(S^1)) has infinitely many components!  

I´ll report on recent work and new tools to understand Hom(G, Homeo+(S^1)).  In particular, I use dynamical methods to give a new lower bound on the number of its components, and show that certain «geometric» actions of G are surprisingly rigid.   

Title: Zero Lebesgue measure for exceptional minimal sets on the circle.

Abstract: I will explain the ideas of proof of the real-analytic case of a longstanding conjecture of Ghys and Sullivan: Every Cantor set that is a minimal-invariant set for the action of a finitely-generated group of circle diffeomorphisms has zero Lebesgue measure. I will also discuss the minimal case, where we prove ergodicity with respect to the Lebesgue measure provided the underlying group is algebraically free. This is joint work with B.Deroin and V.Kleptsyn.