Linear representations of hyperelliptic mapping class groups

TL;DR

This paper investigates linear representations of hyperelliptic mapping class groups derived from finite covers, demonstrating the existence of nontrivial finite orbits in these representations for genus g ≥ 2.

Contribution

It extends previous results on virtual linear representations to hyperelliptic mapping class groups and constructs new representations with finite orbits.

Findings

Existence of virtual linear representations with nontrivial finite orbits for all g ≥ 2.

Extension of results from mapping class groups to hyperelliptic subgroups.

Representations associated to G-coverings ramified over Weierstrass points.

Abstract

Let $p:S\to S_g$ be a finite $G$-covering of a closed surface of genus $g\geq 1$ and let $B$ its branch locus. To this data, it is associated a representation of a finite index subgroup of the mapping class group $\operatorname{Mod}(S_g\smallsetminus B)$ in the centralizer of the group $G$ in the symplectic group $\operatorname{Sp}(H_1(S,{\mathbb Q}))$. They are called \emph{virtual linear representations} of the mapping class group and are related, via a conjecture of Putman and Wieland, to a question of Kirby and Ivanov on the abelianization of finite index subgroup of the mapping class group. The purpose of this paper is to study the restriction of such representations to the hyperelliptic mapping class group $\operatorname{Mod}(S_g,B)^\iota$, which is a subgroup of $\operatorname{Mod}(S_g\smallsetminus B)$ associated to a given hyperelliptic involution $\iota$ on $S_g$. We extend to hyperelliptic mapping class groups some previous results on virtual linear representations of the mapping class group. We then show that, for all $g\geq 2$, there are virtual linear representations of the hyperelliptic mapping class group with nontrivial finite orbits, associated to $G$-coverings of $(S_g,\iota)$ ramified over the locus of Weierstrass points.