  In the preceding paper, Calegari and Dunfield exhibit a sequence of
hyperbolic 3-manifolds which have increasing injectivity radius, and which,
subject to some conjectures in number theory, are rational homology spheres. We
prove unconditionally that these manifolds are rational homology spheres, and
give a sufficient condition for a tower of hyperbolic 3-manifolds to have first
Betti number 0 at each level. The methods involved are purely pro-p group
theoretical.
