Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $\pi$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $\Sigma$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $\Sigma$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.