Duke Mathematical Journal

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

Pierre Albin, Frédéric Rochon, and David Sher

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We establish a Cheeger–Müller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.

Article information

Source
Duke Math. J., Volume 167, Number 10 (2018), 1883-1950.

Dates
Received: 22 January 2015
Revised: 10 August 2017
First available in Project Euclid: 7 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1528358417

Digital Object Identifier
doi:10.1215/00127094-2018-0009

Mathematical Reviews number (MathSciNet)
MR3827813

Zentralblatt MATH identifier
06928113

Subjects
Primary: 58J52: Determinants and determinant bundles, analytic torsion
Secondary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 58J35: Heat and other parabolic equation methods 55N25: Homology with local coefficients, equivariant cohomology 55N33: Intersection homology and cohomology

Keywords
analytic torsion Reidemeister torsion locally symmetric spaces hyperbolic cusps analytic surgery determinant of the Laplacian

Citation

Albin, Pierre; Rochon, Frédéric; Sher, David. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Duke Math. J. 167 (2018), no. 10, 1883--1950. doi:10.1215/00127094-2018-0009. https://projecteuclid.org/euclid.dmj/1528358417


Export citation

References

  • [1] P. Albin, Renormalizing curvature integrals on Poincaré-Einstein manifolds, Adv. Math. 221 (2009), 140–169.
  • [2] P. Albin and F. Rochon, Some index formulae on the moduli space of stable parabolic vector bundles, J. Aust. Math. Soc. 94 (2013), 1–37.
  • [3] P. Albin, F. Rochon, and D. Sher, Resolvent, heat kernel, and torsion under degeneration to fibered cusps, to appear in Mem. Amer. Math. Soc., preprint, arXiv:1410.8406v4 [math.DG].
  • [4] N. Bergeron, M. H. Sengün, and A. Venkatesh, Torsion homology growth and cycle complexity of arithmetic manifolds, Duke Math. J. 165 (2016), 1629–1693.
  • [5] N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), 391–447.
  • [6] J.-M. Bismut, X. Ma, and W. Zhang, Asymptotic torsion and Toeplitz operators, J. Inst. Math. Jussieu 16 (2017), 223–349.
  • [7] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller, with an appendix by F. Laudenbach, Astérisque 205, Soc. Math. France, Paris, 1992.
  • [8] A. Borel and N. R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Stud. 94, Princeton Univ. Press, Princeton, 1980.
  • [9] M. Burger, Asymptotics of small eigenvalues of Riemann surfaces, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 39–40.
  • [10] F. Calegari and A. Venkatesh, A torsion Jacquet-Langlands correspondence, preprint, arXiv:1212.3847v1 [math.NT].
  • [11] J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259–322.
  • [12] X. Dai and X. Huang, The intersection R-torsion for finite cone, preprint, arXiv:1410.6110v2 [math.DG].
  • [13] A. Dar, Intersection $R$-torsion and analytic torsion for pseudomanifolds, Math. Z. 194 (1987), 193–216.
  • [14] J. Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), 437–443.
  • [15] D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (1986), 523–540.
  • [16] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135–162.
  • [17] M. Goresky and R. MacPherson, Intersection homology, II, Invent. Math. 72 (1983), 77–129.
  • [18] C. Guillarmou and D. A. Sher, Low energy resolvent for the Hodge Laplacian: applications to Riesz transform, Sobolev estimates, and analytic torsion, Int. Math. Res. Not. IMRN 2015, no. 15, 6136–6210.
  • [19] G. Harder, “On the cohomology of discrete arithmetically defined groups” in Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, Bombay, 1975, 129–160.
  • [20] L. Hartmann and M. Spreafico, The analytic torsion of a cone over a sphere, J. Math. Pures Appl. (9) 93 (2010), 408–435.
  • [21] L. Hartmann and M. Spreafico, The analytic torsion of a cone over an odd dimensional manifold, J. Geom. Phys. 61 (2011), 624–657.
  • [22] T. Hausel, E. Hunsicker, and R. Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485–548.
  • [23] M. Lesch, A gluing formula for the analytic torsion on singular spaces, Anal. PDE 6 (2013), 221–256.
  • [24] S. Marshall and W. Müller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. 162 (2013), 863–888.
  • [25] R. Mazzeo and B. Vertman, Analytic torsion on manifolds with edges, Adv. Math. 231 (2012), 1000–1040.
  • [26] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Res. Notes in Math. 4, A. K. Peters, Wellesley, Mass., 1993.
  • [27] R. B. Melrose and V. Nistor, Homology of pseudodifferential operators, I: Manifolds with boundary, preprint, arXiv:funct-an/9606005v2.
  • [28] R. B. Melrose and F. Rochon, Families index for pseudodifferential operators on manifolds with boundary, Int. Math. Res. Not. IMRN 2004, no. 22, 1115–1141.
  • [29] R. B. Melrose and X. Zhu, Resolution of the canonical fiber metric for a Lefschetz fibration, J. Differential Geom. 108 (2018), 295–317.
  • [30] P. Menal-Ferrer and J. Porti, Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol. 7 (2014), 69–119.
  • [31] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.
  • [32] W. Müller, Analytic torsion and $R$-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233–305.
  • [33] W. Müller, Analytic torsion and $R$-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721–753.
  • [34] W. Müller, “The asymptotics of the Ray-Singer analytic torsion for hyperbolic $3$-manifolds” in Metric and Differential Geometry, Progr. Math. 297, Birkhäuser, Basel, 2012, 317–352.
  • [35] W. Müller and J. Pfaff, Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. 263 (2012), 2615–2675.
  • [36] W. Müller and J. Pfaff, Analytic torsion and $L^{2}$-torsion of compact locally symmetric manifolds, J. Differential Geom. 95 (2013), 71–119.
  • [37] W. Müller and J. Pfaff, On the asymptotics of the Ray-Singer analytic torsion for compact hyperbolic manifolds, Int. Math. Res. Not. IMRN 2013, no. 13, 2945–2983.
  • [38] W. Müller and J. Pfaff, The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume, J. Funct. Anal. 267 (2014), 2731–2786.
  • [39] W. Müller and J. Pfaff, On the growth of torsion in the cohomology of arithmetic groups, Math. Ann. 359 (2014), 537–555.
  • [40] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.
  • [41] J. Park, Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps, J. Funct. Anal. 257 (2009), 1713–1758.
  • [42] J. Pfaff, Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z. 277 (2014), 953–974.
  • [43] J. Pfaff, Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups, Ann. Global Anal. Geom. 45 (2014), 267–285.
  • [44] J. Pfaff, Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume, J. Reine Angew. Math. 703 (2015), 115–145.
  • [45] J. Pfaff, A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, J. Inst. Math. Jussieu 16 (2017), 673–743.
  • [46] J. Raimbault, Asymptotics of analytic torsion for hyperbolic three-manifolds, to appear in Comment. Math. Helv., preprint, arXiv:1212.3161v3 [math.DG].
  • [47] J. Raimbault, Analytic, Reidemeister and homological torsion for congruence three-manifolds, preprint, arXiv:1307.2845v2 [math.GT].
  • [48] D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210.
  • [49] R. Seeley and I. M. Singer, Extending $\overline{\partial}$ to singular Riemann surfaces, J. Geom. Phys. 5 (1988), 121–136.
  • [50] D. A. Sher, Conic degeneration and the determinant of the Laplacian, J. Anal. Math. 126 (2015), 175–226.
  • [51] B. Vaillant, Index- and spectral theory for manifolds with generalized fibred cusps, Ph.D. dissertation, Bonner Math. Schriften 344, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2001.
  • [52] B. Vertman, Analytic torsion of a bounded generalized cone, Comm. Math. Phys. 290 (2009), 813–860.
  • [53] B. Vertman, Cheeger-Mueller theorem on manifolds with cusps, preprint, arXiv:1411.0615v3 [math.DG].
  • [54] S. A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), 283–315.
  • [55] S. A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), 417–472.
  • [56] S. A. Wolpert, Families of Riemann Surfaces and Weil-Petersson Geometry, CBMS Reg. Conf. Ser. Math. 113, Amer. Math. Soc., Providence, 2010.