Injectivity radii of hyperbolic integer homology 3spheres
Abstract
We construct hyperbolic integer homology 3spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3manifolds which BenjaminiSchramm converge to H^3 whose normalized RaySinger analytic torsions do not converge to the L^2analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that BenjaminiSchramm 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 3manifolds, and we give experimental results which support this and related conjectures.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.0391
 Bibcode:
 2013arXiv1304.0391B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry;
 Mathematics  Number Theory;
 57M50;
 30F40
 EPrint:
 29 pages, 11 figures. v2: Incorporates referee's comments. To appear in Geometry and Topology