Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

Abstract

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel–Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

    Since \(\frac{x}{\rho }=\sin \theta \) in the coordinates of [ARS14].

References

  1. [ARS14]

    PierreAlbin, FrédéricRochon, and DavidSher, Resolvent, heat kernel, and torsion under degeneration to fibered cusps. arXiv:1410.8406, to appear in Mem. Amer. Math. Soc., (2014).

  2. [ARS18]

    PierreAlbin, FrédéricRochon, and DavidSher, Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Duke Math. J., (10)167 (2018), 1883–1950.

    MathSciNet  Article  Google Scholar 

  3. [BV13]

    NicolasBergeron and AkshayVenkatesh, The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu, (2)12 (2013), 391–447.

    MathSciNet  Article  Google Scholar 

  4. [BW80]

    ABorel and NWallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Stud., vol. 94, Princeton University Press (1980).

  5. [BZ92]

    Jean-MichelBismut and WeipingZhang, An extension of a theorem by Cheeger and Müller. Astérisque, (205) (1992), 235, With an appendix by Francois Laudenbach.

  6. [Che79]

    JeffCheeger, Analytic torsion and the heat equation. Ann. of Math. (2), (2)109 (1979), 259–322.

    MathSciNet  Article  Google Scholar 

  7. [CV12]

    FrankCalegari and AkshayVenkatesh, A torsion Jacquet-Langlands correspondence, available online at arXiv:1212.3847, (2012).

  8. [Fra35]

    WFranz, Über die Torsion einer Überdeckung. J. für die reine und angew. Math., 173 (1935), 245–253.

    MathSciNet  MATH  Google Scholar 

  9. [GW09]

    RoeGoodman and Nolan R.Wallach, Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol. 255, Springer, Dordrecht, (2009).

    Google Scholar 

  10. [Har75]

    GHarder, On the cohomology of discrete arithmetically defined groups, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, Bombay, pp. 129–160, (1975)

  11. [HHM04]

    TamásHausel, EugenieHunsicker, and RafeMazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. (3)122 (2004), 485–548.

    MathSciNet  Article  Google Scholar 

  12. [Kos61]

    BertramKostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2), 74 (1961), 329–387.

    MathSciNet  Article  Google Scholar 

  13. [Les13]

    MatthiasLesch, A gluing formula for the analytic torsion on singular spaces, Anal. PDE, (1)6 (2013), 221–256.

    MathSciNet  Article  Google Scholar 

  14. [Mel93]

    Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters Ltd., Wellesley, MA, (1993)

  15. [MFP12]

    PereMenal-Ferrer and JoanPorti, Twisted cohomology for hyperbolic three manifolds. Osaka J. Math., (3)49 (2012), 741–769.

    MathSciNet  MATH  Google Scholar 

  16. [MFP14]

    PereMenal-Ferrer and JoanPorti, Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol. (1)7 (2014), 69–119.

    MathSciNet  Article  Google Scholar 

  17. [Mil66]

    J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.

    MathSciNet  Article  Google Scholar 

  18. [MM63]

    YozôMatsushima and ShingoMurakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2), 78 (1963), 365–416.

    MathSciNet  Article  Google Scholar 

  19. [MM95]

    RafeMazzeo and Richard B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal., (1)5 (1995), 14–75.

    MathSciNet  Article  Google Scholar 

  20. [MM13]

    SimonMarshall and WernerMüller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. (5)(2013), 863–888.

    MathSciNet  Article  Google Scholar 

  21. [MP12]

    WernerMüller and JonathanPfaff, Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. (9)263 (2012), 2615–2675.

    MathSciNet  Article  Google Scholar 

  22. [MP14]

    WernerMüller and JonathanPfaff, On the growth of torsion in the cohomology of arithmetic groups, Math. Ann., (1-2)359 (2014), 537–555.

    MathSciNet  Article  Google Scholar 

  23. [MR19]

    WMüller and FRochon, Exponential growth of torsion for sequences of hyperbolic manifolds of finite volume, posted on arXiv, (2019).

  24. [Mül78]

    WernerMüller, Analytic torsion and \(R\)-torsion of Riemannian manifolds, Adv. in Math., (3)28 (1978), 233–305.

    MathSciNet  Article  Google Scholar 

  25. [Mül93]

    WernerMüller, Analytic torsion and \(R\)-torsion for unimodular representations, J. Amer. Math. Soc., (3)6 (1993), 721–753.

    MathSciNet  Article  Google Scholar 

  26. [Mül12]

    WernerMüller, The asymptotics of the Ray-Singer analytic torsion for hyperbolic \(3\)-manifolds, Metric and Dinferential Geometry. The Jeff Cheeger Anniversary Volume, Progress in Math., vol. 297, Birkhäuser, pp. 317–352, (2012)

  27. [Oni04]

    Arkady L. Onishchik, Lectures on real semisimple Lie algebras and their representations, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, (2004).

  28. [Pfa14]

    JonathanPfaff, Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z., (3-4)277 (2014), 953–974.

    MathSciNet  Article  Google Scholar 

  29. [Pfa17]

    JonathanPfaff, A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, J. Inst. Math. Jussieu (4)16 (2017), 673–743.

    MathSciNet  Article  Google Scholar 

  30. [PR20]

    JPfaff and JRaimbault, On the torsion in symmetric powers on congruence subgroups of Bianchi groups, Trans. AMS, (1)373 (2020), 109–148.

    MathSciNet  Article  Google Scholar 

  31. [Rei35]

    KurtReidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg, (1)11 (1935), 102–109.

    MathSciNet  Article  Google Scholar 

  32. [RS71]

    DB. Ray and IM. Singer, \(R\)-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210.

    MathSciNet  Article  Google Scholar 

  33. [Shu01]

    MA. Shubin, Pseudodifferential operators and spectral theory, second ed., Springer-Verlag, Berlin, Translated from the 1978 Russian original by Stig I. Andersson. (2001)

  34. [vE58]

    WT. van Est, A generalization of the Cartan-Leray spectral sequence. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math. 20 (1958), 399–413.

    MathSciNet  Article  Google Scholar 

  35. [Zuc83]

    StevenZucker, \(L_{2}\) cohomology of warped products and arithmetic groups, Invent. Math., (2)70 (1982/83), 169–218.

Download references

Acknowledgements

The authors are grateful to the hospitality of the Centre International de Rencontres Mathématiques (CIRM) where this project started. The second author was supported by NSERC and a Canada Research Chair.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Frédéric Rochon.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Müller, W., Rochon, F. Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps. Geom. Funct. Anal. (2020). https://doi.org/10.1007/s00039-020-00536-2

Download citation