Injectivity radii of hyperbolic integer homology 3–spheres

Brock, Jeffrey F¹ ; Dunfield, Nathan M²

¹ Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA
² Department of Mathematics, University of Illinois, 1409 W Green St, Urbana, IL 61801, USA

Geometry & topology, Tome 19 (2015) no. 1, pp. 497-523.

Résumé

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 ${ℍ}^3$ whose normalized Ray–Singer analytic torsions do not converge to the $L^2$–analytic torsion of ${ℍ}^3$. 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.

Détail

DOI : 10.2140/gt.2015.19.497

Classification : 57M50, 30F40
Keywords: hyperbolic integer homology sphere, injectivity radius, torsion growth, Ray–Singer analytic torsion, Benjamini–Schramm convergence

Bibliographie

[1] M Abert, N Bergeron, I Biringer, T Gelander, N Nikolov, J Raimbault, I Samet, On the growth of L2–invariants for sequences of lattices in Lie groups,

[2] I Agol, Criteria for virtual fibering, J. Topol. 1 (2008) 269

[3] N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013) 391

[4] W Bosma, J Cannon, C Fieker, A Steel, Handbook of MAGMA functions

[5] N Boston, J S Ellenberg, Pro-p groups and towers of rational homology spheres, Geom. Topol. 10 (2006) 331

[6] J F Brock, K W Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004) 33

[7] J F Brock, Y N Minsky, H Namazi, J Souto, Bounded combinatorics and uniform models for hyperbolic 3–manifolds,

[8] F Calegari, N M Dunfield, Data files and Magma programs for ‘Automorphic forms and rational homology 3–spheres’

[9] F Calegari, N M Dunfield, Automorphic forms and rational homology 3–spheres, Geom. Topol. 10 (2006) 295

[10] F Calegari, A Venkatesh, A torsion Jacquet–Langlands correspondence,

[11] J Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979) 259

[12] V Gadre, The limit set of the handlebody set has measure zero

[13] S P Kerckhoff, The measure of the limit set of the handlebody group, Topology 29 (1990) 27

[14] T T Q Le, Hyperbolic volume, Mahler measure and homology growth

[15] D D Long, A W Reid, Simple quotients of hyperbolic 3–manifold groups, Proc. Amer. Math. Soc. 126 (1998) 877

[16] J Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992) 471

[17] W Lück, Approximating L2–invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455

[18] C T Mcmullen, Renormalization and 3–manifolds which fiber over the circle, 142, Princeton Univ. Press (1996)

[19] Y N Minsky, Bounded geometry for Kleinian groups, Invent. Math. 146 (2001) 143

[20] W Müller, Analytic torsion and R–torsion of Riemannian manifolds, Adv. in Math. 28 (1978) 233

[21] H Namazi, Heegaard splittings and hyperbolic geometry, PhD thesis, Yale University (2005)

[22] H Namazi, J Souto, Heegaard splittings and pseudo-Anosov maps, Geom. Funct. Anal. 19 (2009) 1195

[23] A Page, Algorithms for arithmetic Kleinian groups

[24] A Page, KleinianGroups, version 1.0

[25] C Petronio, J Porti, Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem, Expo. Math. 18 (2000) 1

[26] J S Purcell, J Souto, Geometric limits of knot complements, J. Topol. 3 (2010) 759

[27] M H Şengün, On the integral cohomology of Bianchi groups, Exp. Math. 20 (2011) 487

[28] M H Şengün, On the integral cohomology of Bianchi groups (2012)

[29] M H Şengün, On the torsion homology of nonarithmetic hyperbolic tetrahedral groups, Int. J. Number Theory 8 (2012) 311

[30] W P Thurston, Hyperbolic structures on 3–manifolds, II : Surface groups and 3–manifolds which fiber over the circle,

[31] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)

[32] G Tian, A pinching theorem on manifolds with negative curvature, preprint

[33] S K Yeung, Betti numbers on a tower of coverings, Duke Math. J. 73 (1994) 201