Prym Representations of the Handlebody Group

TL;DR

This paper studies Prym representations of the handlebody group, focusing on their images in the cyclic case, and explores how these representations relate to the subgroup of twists about meridians.

Contribution

It extends the understanding of Prym representations by analyzing their images when restricted to the handlebody and twist groups, especially in the cyclic case.

Findings

Determined the image of Prym representations for cyclic groups.

Analyzed the restriction of Prym representations to the handlebody group.

Provided new insights into the structure of the twist subgroup.

Abstract

Let $S$ be an oriented, closed surface of genus $g.$ The mapping class group of $S$ is the group of orientation preserving homeomorphisms of $S$ modulo isotopy. In 1997, Looijenga introduced the Prym representations, which are virtual representations of the mapping class group that depend on a finite, abelian group. Let $V$ be a genus $g$ handlebody with boundary $S$. The handlebody group is the subgroup of those mapping classes of $S$ that extend over $V.$ The twist group is the subgroup of the handlebody group generated by twists about meridians. Here, we restrict the Prym representations to the handlebody group and further to the twist group. We determine the image of the representations in the cyclic case.