Pseudo-Anosov action on the $SU(2)$-character variety of $S_2$

TL;DR

This paper investigates the dynamics of certain mapping class subgroups acting on the $SU(2)$-character variety of a genus 2 surface, revealing the existence of subgroups with pseudo-Anosov elements that preserve rational functions.

Contribution

It demonstrates the existence of a genus 2 mapping class subgroup with infinitely many pseudo-Anosov elements that admit invariant rational functions on the $SU(2)$-character variety.

Findings

Existence of a subgroup with infinitely many pseudo-Anosov elements

Presence of invariant rational functions under subgroup action

Extension of previous studies on mapping class group actions

Abstract

In this article, we continue the study of the action of subgroups of the mapping class group on the $SU(2)$-character variety. We prove the existence of a mapping class subgroup on the surface of genus $2$, containing infinitely many pseudo-Anosov elements, which admit an invariant rational function on $SU(2)$-character variety of $S_2$.