  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.
