  Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of
genus g. This is the subgroup of the mapping class group of S_g consisting of
elements that act trivially on H_1(S_g;Z) and that commute with some fixed
hyperelliptic involution of S_g. We prove that the cohomological dimension of
SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely
generated when g > 1. In particular, SI(S_3) is not finitely presentable.
Finally, we apply our main results to show that the kernel of the Burau
representation of the braid group B_n at t = -1 has cohomological dimension
equal to the integer part of n/2, and it has infinitely generated homology in
this top dimension.
