Transitivity of normal subgroups of the mapping class group on character varieties

TL;DR

This paper proves that non-trivial normal subgroups of the mapping class group act almost minimally on the character variety of a surface, with almost every orbit being dense, revealing deep dynamical properties of these group actions.

Contribution

It establishes the almost minimal action of non-trivial normal subgroups on the character variety of surfaces, a new insight into the dynamics of mapping class group actions.

Findings

Almost every orbit under the subgroup action is dense.

Normal subgroups act with almost minimal dynamics on the character variety.

The result applies to surfaces of genus at least 2.

Abstract

We prove that the action of any non-trivial normal subgroup of the mapping class group of a surface of genus $g\geqslant 2$ is almost minimal on the character variety $X(\pi_1\Sigma_g,{\rm SU}_2)$: the orbit of almost every point is dense.