Factoring in the hyperelliptic Torelli group

TL;DR

This paper develops an algorithmic method to factor elements of the hyperelliptic Torelli group into Dehn twists, enhancing understanding of its structure and providing tools for explicit computations.

Contribution

It introduces a novel algorithmic approach for factoring elements in the hyperelliptic Torelli group into generating Dehn twists, building on previous structural results.

Findings

Successfully factors a broad class of elements into Dehn twists

Demonstrates the effectiveness of the algorithm on basic elements

Provides a computational tool for studying the hyperelliptic Torelli group

Abstract

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic elements.