#### Volume 19, issue 1 (2015)

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Injectivity radii of hyperbolic integer homology $3$–spheres

### Jeffrey F Brock and Nathan M Dunfield

Geometry & Topology 19 (2015) 497–523
##### 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 that 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.

##### Keywords
hyperbolic integer homology sphere, injectivity radius, torsion growth, Ray–Singer analytic torsion, Benjamini–Schramm convergence
Primary: 57M50
Secondary: 30F40