Geometry & Topology

Injectivity radii of hyperbolic integer homology $3$–spheres

Jeffrey F Brock and Nathan M Dunfield

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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 3 whose normalized Ray–Singer analytic torsions do not converge to the L2–analytic torsion of 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.

Article information

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

Received: 4 February 2014
Revised: 7 March 2014
Accepted: 20 May 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F40: Kleinian groups [See also 20H10]

hyperbolic integer homology sphere injectivity radius torsion growth Ray–Singer analytic torsion Benjamini–Schramm convergence


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.

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