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Injectivity radii of
hyperbolic integer homology $3$–spheres
Jeffrey F Brock and Nathan M Dunfield |
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Geometry & Topology 19 (2015) 497–523
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Abstract |
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We construct hyperbolic integer homology –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 –manifolds that Benjamini–Schramm converge to whose normalized Ray–Singer analytic torsions do not converge to the –analytic torsion of . 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 –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
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Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 30F40
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References |
Publication
Received: 4 February 2014
Revised: 7 March 2014
Accepted: 20 May 2014
Published: 27 February 2015
Proposed: Ian Agol
Seconded: John Lott, Dmitri Burago
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Authors |
| Jeffrey F Brock | |
| Department of Mathematics Brown University Box 1917 Providence, RI 02912 USA |
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| http://www.math.brown.edu/~brock/ | |
| Nathan M Dunfield | |
| Department of Mathematics University of Illinois 1409 W Green St Urbana, IL 61801 USA |
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| http://dunfield.info | |
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