The mapping class group action on SU(3)-character varieties

TL;DR

This paper proves that the mapping class group acts ergodically on the SU(3)-character variety of a genus one surface with one boundary, with respect to the natural symplectic measure, revealing deep dynamical properties.

Contribution

It establishes the ergodicity of the mapping class group action on SU(3)-character varieties for a specific surface, a novel result in the study of character varieties.

Findings

The action is ergodic with respect to the symplectic measure.

The result applies to genus one surfaces with one boundary component.

It advances understanding of the dynamics of surface group representations.

Abstract

Let $\Sigma$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\Gamma$ of $\Sigma$ acts on the SU(3)-character variety of $\Sigma$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.