Geometry & Topology

Injectivity radii of hyperbolic integer homology $3$–spheres

Jeffrey F Brock and Nathan M Dunfield

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858687

Digital Object Identifier
doi:10.2140/gt.2015.19.497

Mathematical Reviews number (MathSciNet)
MR3318758

Zentralblatt MATH identifier
1312.57022

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

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

Citation

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. https://projecteuclid.org/euclid.gt/1510858687


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References

  • M Abert, N Bergeron, I Biringer, T Gelander, N Nikolov, J Raimbault, I Samet, On the growth of $L^2$–invariants for sequences of lattices in Lie groups
  • I Agol, Criteria for virtual fibering, J. Topol. 1 (2008) 269–284
  • N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013) 391–447
  • W Bosma, J Cannon, C Fieker, A Steel, Handbook of MAGMA functions Available at \setbox0\makeatletter\@url http://magma.maths.usyd.edu.au/ {\unhbox0
  • N Boston, J S Ellenberg, Pro-$p$ groups and towers of rational homology spheres, Geom. Topol. 10 (2006) 331–334
  • J F Brock, K W Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004) 33–93
  • J F Brock, Y N Minsky, H Namazi, J Souto, Bounded combinatorics and uniform models for hyperbolic $3$–manifolds
  • F Calegari, N M Dunfield, Data files and Magma programs for `Automorphic forms and rational homology $3$–spheres' Available at \setbox0\makeatletter\@url http://www.computop.org/software/twist-congruence/ {\unhbox0
  • F Calegari, N M Dunfield, Automorphic forms and rational homology $3$–spheres, Geom. Topol. 10 (2006) 295–329
  • F Calegari, A Venkatesh, A torsion Jacquet–Langlands correspondence
  • J Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979) 259–322
  • V Gadre, The limit set of the handlebody set has measure zero Appendix to “Are large distance Heegaard splittings generic?” by Lustig and Moriah, J. Reine Angew. Math. 670 (2012) 115–119
  • S P Kerckhoff, The measure of the limit set of the handlebody group, Topology 29 (1990) 27–40
  • T T Q Le, Hyperbolic volume, Mahler measure and homology growth Available at \setbox0\makeatletter\@url http://www.math.columbia.edu/~volconf09/notes/leconf.pdf {\unhbox0
  • D D Long, A W Reid, Simple quotients of hyperbolic $3$–manifold groups, Proc. Amer. Math. Soc. 126 (1998) 877–880
  • J Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992) 471–510
  • W Lück, Approximating $L^2$–invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455–481
  • C T McMullen, Renormalization and $3$–manifolds which fiber over the circle, Annals of Math. Studies 142, Princeton Univ. Press (1996)
  • Y N Minsky, Bounded geometry for Kleinian groups, Invent. Math. 146 (2001) 143–192
  • W Müller, Analytic torsion and $R$–torsion of Riemannian manifolds, Adv. in Math. 28 (1978) 233–305
  • H Namazi, Heegaard splittings and hyperbolic geometry, PhD thesis, Yale University (2005) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305372837 {\unhbox0
  • H Namazi, J Souto, Heegaard splittings and pseudo-Anosov maps, Geom. Funct. Anal. 19 (2009) 1195–1228
  • A Page, Algorithms for arithmetic Kleinian groups Available at \setbox0\makeatletter\@url http://www.birs.ca/events/2012/5-day-workshops/12w5075/videos {\unhbox0
  • A Page, KleinianGroups, version $1.0$ Available at \setbox0\makeatletter\@url http://www.normalesup.org/~page/Recherche/Logiciels/logiciels-en.html {\unhbox0
  • C Petronio, J Porti, Negatively oriented ideal triangulations and a proof of Thurston's hyperbolic Dehn filling theorem, Expo. Math. 18 (2000) 1–35
  • J S Purcell, J Souto, Geometric limits of knot complements, J. Topol. 3 (2010) 759–785
  • M H Şengün, On the integral cohomology of Bianchi groups, Exp. Math. 20 (2011) 487–505
  • M H Şengün, On the integral cohomology of Bianchi groups (2012) video talk at Torsion in homology of arithmetic groups
  • M H Şengün, On the torsion homology of nonarithmetic hyperbolic tetrahedral groups, Int. J. Number Theory 8 (2012) 311–320
  • W P Thurston, Hyperbolic structures on $3$–manifolds, II: Surface groups and $3$–manifolds which fiber over the circle
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
  • G Tian, A pinching theorem on manifolds with negative curvature, preprint
  • S-K Yeung, Betti numbers on a tower of coverings, Duke Math. J. 73 (1994) 201–226