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

*(English)*Zbl 1408.58025Authors’ abstract: We established 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 the define Analytic torsion, and we relate it to the intersection \(R\)-torsion of A. Dar [Math. Z. 194, 193–216 (1987; Zbl 0605.57013)] 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 [the authors, “Resolvent, heat kernel and torsion under degeneration to fibered cusps”, Preprint, arXiv:1410.8406v4].

Reviewer: Luiz Hartmann (São Carlos)

##### MSC:

58J52 | Determinants and determinant bundles, analytic torsion |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

55N25 | Homology with local coefficients, equivariant cohomology |

55N33 | Intersection homology and cohomology in algebraic topology |

##### Keywords:

manifolds with cusps; Cheeger-Müller theorem; intersection \(R\)-torsion; analytic torsion; locally symmetric spaces; renormalized trace; Reidemeister torsion; hyperbolic cusps; analytic surgery; determinant of the Laplacian##### Citations:

Zbl 0605.57013
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\textit{P. Albin} et al., Duke Math. J. 167, No. 10, 1883--1950 (2018; Zbl 1408.58025)

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##### References:

[1] | P. Albin, Renormalizing curvature integrals on Poincaré-Einstein manifolds, Adv. Math. 221 (2009), 140-169. · Zbl 1170.53017 |

[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. · Zbl 1318.58011 |

[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. · Zbl 1351.11031 |

[5] | N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), 391-447. · Zbl 1266.22013 |

[6] | J.-M. Bismut, X. Ma, and W. Zhang, Asymptotic torsion and Toeplitz operators, J. Inst. Math. Jussieu 16 (2017), 223-349. · Zbl 1381.58015 |

[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. · Zbl 0443.22010 |

[9] | M. Burger, Asymptotics of small eigenvalues of Riemann surfaces, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 39-40. · Zbl 0642.58010 |

[10] | F. Calegari and A. Venkatesh, A torsion Jacquet-Langlands correspondence, preprint, arXiv:1212.3847v1 [math.NT]. · Zbl 1468.11002 |

[11] | J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259-322. · Zbl 0412.58026 |

[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. · Zbl 0605.57012 |

[14] | J. Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), 437-443. · Zbl 0502.58038 |

[15] | D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (1986), 523-540. · Zbl 0621.53035 |

[16] | M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135-162. · Zbl 0448.55004 |

[17] | M. Goresky and R. MacPherson, Intersection homology, II, Invent. Math. 72 (1983), 77-129. · Zbl 0529.55007 |

[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. · Zbl 1331.58026 |

[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. · Zbl 1222.58026 |

[21] | L. Hartmann and M. Spreafico, The analytic torsion of a cone over an odd dimensional manifold, J. Geom. Phys. 61 (2011), 624-657. · Zbl 1222.58027 |

[22] | T. Hausel, E. Hunsicker, and R. Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485-548. · Zbl 1062.58002 |

[23] | M. Lesch, A gluing formula for the analytic torsion on singular spaces, Anal. PDE 6 (2013), 221-256. · Zbl 1276.58010 |

[24] | S. Marshall and W. Müller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. 162 (2013), 863-888. · Zbl 1316.11042 |

[25] | R. Mazzeo and B. Vertman, Analytic torsion on manifolds with edges, Adv. Math. 231 (2012), 1000-1040. · Zbl 1255.58012 |

[26] | R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Res. Notes in Math. 4, A. K. Peters, Wellesley, Mass., 1993. · Zbl 0796.58050 |

[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. · Zbl 1086.58011 |

[29] | R. B. Melrose and X. Zhu, Resolution of the canonical fiber metric for a Lefschetz fibration, J. Differential Geom. 108 (2018), 295-317. · Zbl 1404.30047 |

[30] | P. Menal-Ferrer and J. Porti, Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol. 7 (2014), 69-119. · Zbl 1302.57044 |

[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. · Zbl 1264.58026 |

[35] | W. Müller and J. Pfaff, Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. 263 (2012), 2615-2675. · Zbl 1277.58018 |

[36] | W. Müller and J. Pfaff, Analytic torsion and \(L^{2}\)-torsion of compact locally symmetric manifolds, J. Differential Geom. 95 (2013), 71-119. · Zbl 1281.58022 |

[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. · Zbl 1323.58024 |

[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. · Zbl 1303.58013 |

[39] | W. Müller and J. Pfaff, On the growth of torsion in the cohomology of arithmetic groups, Math. Ann. 359 (2014), 537-555. · Zbl 1318.11072 |

[40] | B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. · Zbl 0653.53022 |

[41] | J. Park, Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps, J. Funct. Anal. 257 (2009), 1713-1758. · Zbl 1181.58023 |

[42] | J. Pfaff, Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z. 277 (2014), 953-974. · Zbl 1298.58021 |

[43] | J. Pfaff, Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups, Ann. Global Anal. Geom. 45 (2014), 267-285. · Zbl 1323.11031 |

[44] | J. Pfaff, Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume, J. Reine Angew. Math. 703 (2015), 115-145. · Zbl 1323.58023 |

[45] | J. Pfaff, A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, J. Inst. Math. Jussieu 16 (2017), 673-743. · Zbl 1378.58030 |

[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. · Zbl 0239.58014 |

[49] | R. Seeley and I. M. Singer, Extending \(\overline{∂}\) to singular Riemann surfaces, J. Geom. Phys. 5 (1988), 121-136. · Zbl 0692.30038 |

[50] | D. A. Sher, Conic degeneration and the determinant of the Laplacian, J. Anal. Math. 126 (2015), 175-226. · Zbl 1328.58031 |

[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. · Zbl 1059.58018 |

[52] | B. Vertman, Analytic torsion of a bounded generalized cone, Comm. Math. Phys. 290 (2009), 813-860. · Zbl 1210.58026 |

[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. · Zbl 0629.58029 |

[55] | S. A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), 417-472. · Zbl 0698.53002 |

[56] | S. A. Wolpert, Families of Riemann Surfaces and Weil-Petersson Geometry, CBMS Reg. Conf. Ser. Math. 113, Amer. Math. Soc., Providence, 2010. · Zbl 1198.30049 |

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