Ghost polygons, Poisson bracket and convexity

The moduli space of Anosov representations of a surface group in a group $\mathsf G$, which is an open set in the character variety, admits many more natural functions than the regular functions: length functions, correlation functions. We  compute the Poisson bracket of those functions using some combinatorial device, show that the set of those functions is stable under the Poisson bracket and give an application to the convexity of length functions, generalizing the result of Kerckhoff on Teichmüller space.

We shall start by giving an introduction to Anosov representations, define precisely what are the functions we consider and explain the combinatorial device involved.

This is a joint work with Martin Bridgeman.