Injectivity radii of hyperbolic integer homology 3-spheres

Abstract

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.

29 pages, 11 figures. v2: Incorporates referee's comments. To appear in Geometry and Topology

References in corpus (3)

• Criteria for virtual fibering
• Pro-p groups and towers of rational homology spheres
• Geometric limits of knot complements

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