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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

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.

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hyperbolic integer homology sphere, injectivity radius, torsion growth, Ray–Singer analytic torsion, Benjamini–Schramm convergence
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 30F40
Received: 4 February 2014
Revised: 7 March 2014
Accepted: 20 May 2014
Published: 27 February 2015
Proposed: Ian Agol
Seconded: John Lott, Dmitri Burago
Jeffrey F Brock
Department of Mathematics
Brown University
Box 1917
Providence, RI 02912
Nathan M Dunfield
Department of Mathematics
University of Illinois
1409 W Green St
Urbana, IL 61801