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.
ASJC Scopus subject areas
- Geometry and Topology