## Geometry & Topology

### 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 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.

#### Article information

Source
Geom. Topol., Volume 19, Number 1 (2015), 497-523.

Dates
Revised: 7 March 2014
Accepted: 20 May 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858687

Digital Object Identifier
doi:10.2140/gt.2015.19.497

Mathematical Reviews number (MathSciNet)
MR3318758

Zentralblatt MATH identifier
1312.57022

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

Brock, Jeffrey F; Dunfield, Nathan M. Injectivity radii of hyperbolic integer homology $3$–spheres. Geom. Topol. 19 (2015), no. 1, 497--523. doi:10.2140/gt.2015.19.497. https://projecteuclid.org/euclid.gt/1510858687

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