Three proofs from dynamics of rigidity of surface group actions
Kathryn Mann (University of California, Berkeley)
MSRI: Simons Auditorium
In previous talks (not a prerequisite!), I've described examples of actions of a surface group G on the circle that are totally rigid -- they are essentially isolated points in the representation space Hom(G, Homeo+(S^1))/~. These examples are interesting from many perspectives, ranging from foliation theory to the classification of connected components of representation spaces.
In this talk, I will illustrate three separate approaches to prove rigidity of these actions, including my original proof. Each one uses fundamentally different techniques, but all have a common dynamical flavor:
1. Structural stability of Anosov foliations (Ghys/Bowden, under extra hypotheses)
2. Rotation number "trace coordinates" on the representation space (Mann)
3. New "ping-pong" lemmas for surface groups (Matsumoto)
A symmetric space is Hermitian if it admits a complex structure preserved by the isometry group. In this introductory talk I will describe various geometric features of these spaces. I will focus particularly on some natural notion of boundary arising in this context, and emphasize the role played by these objects in studying rigidity questions.