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

TL;DR

This paper investigates the topology and symplectic structure of a space of six-point configurations in the hyperbolic plane, analyzing the mapping class group's action to classify ergodic components of the genus 2 surface's character variety.

Contribution

It provides a detailed study of the topology, symplectic structure, and mapping class group action on the space related to genus 2 surface character varieties, completing previous classifications.

Findings

Topology and symplectic structure of the space are characterized.

Mapping class group action on the space is described.

Classification of ergodic components in genus 2 character variety is completed.

Abstract

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.