Injectivity radii of hyperbolic integer homology 3-spheres

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to the L^2-analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures.