The modular action on PSL2(R)-characters in genus 2

Abstract

We explore the dynamics of the action of the mapping class group in genus $2$ on the ${PSL}_2\left(\mathbb{R}\right)$-character variety. We prove that this action is ergodic on the connected components of Euler class $\pm 1$, as it was conjectured by Goldman. In the connected component of Euler class $0$ there are two invariant open subsets; on one of them the action is ergodic. In this process we give a partial answer to a question posed by Bowditch.