Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1

TL;DR

This paper proves that the hyperelliptic Torelli group is generated by specific Dehn twists, confirming a conjecture, and explores implications for the structure of Torelli space and the kernel of the Burau representation at t=-1.

Contribution

It establishes a generating set for the hyperelliptic Torelli group and related kernels, confirming Hain's conjecture and linking algebraic and geometric structures.

Findings

Hyperelliptic Torelli group generated by Dehn twists about separating curves

Kernel of Burau representation at t=-1 generated similarly

Adding curves of compact type makes Torelli space simply connected

Abstract

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.