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

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.

37 pages, 9 figures; Major revision, to appear in Inventiones Mathematicae

References in corpus (2)

• Image of the Burau Representation at $d$-th Roots of unity
• Factoring in the hyperelliptic Torelli group

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• Problems, Questions, and Conjectures about Mapping Class Groups
• The liftable mapping class group of balanced superelliptic covers
• Factoring in the hyperelliptic Torelli group
• Geometric normal subgroups in mapping class groups of punctured surfaces
• Hyperelliptic graphs and the period mapping on outer space
• Linear representations of hyperelliptic mapping class groups
• Braid groups and symplectic Steinberg groups
• Meanders, hyperelliptic pillowcase covers, and the Johnson filtration