On power subgroups of Dehn twists in hyperelliptic mapping class groups

TL;DR

This paper investigates the index of power subgroups generated by Dehn twists in hyperelliptic mapping class groups and punctured sphere groups, revealing conditions under which these subgroups have infinite index using skein module representations.

Contribution

It proves the infinite index of certain power subgroups in hyperelliptic and punctured sphere mapping class groups, completing cases left open in prior research.

Findings

Normal closure of fifth power of a half-twist has infinite index in $ ext{Mod}(0,2n)$.

Subgroups generated by $m$-th powers of Dehn twists have infinite index for $m eq 6$ in hyperelliptic groups.

Uses projective representations from Kauffman bracket skein modules to analyze subgroup indices.

Abstract

This paper contains two topics, the index of a power subgroup in the mapping class group $\mathcal{M}(0,2n)$ of a $2n$-punctured sphere and in the hyperelliptic mapping class group $\Delta(g,0)$ of an oriented closed surface of genus $g$. The main tool is a projective representation of $\mathcal{M}(0,2n)$ obtained through the Kauffman bracket skein module. For $\mathcal{M}(0,2n)$, we prove that the normal closure of the fifth power of a half-twist has infinite index. This is the remaining case of a Masbaum's work. For $\Delta(g,0)$, we consider the normal closure of $m$-th powers of Dehn twists along all symmetric simple closed curves. We show the subgroup has infinite index if $m\geq 5$ and $m\neq 6$ for any $g\geq 2$.