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 that Benjamini–Schramm converge to ℍ³ whose normalized Ray–Singer analytic torsions do not converge to the L² –analytic torsion of ℍ³. 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.