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Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists


We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

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  1. The case of even dimension is trivial [Sh19, Remark 5.12] (c.f. Remark 4.5).

  2. By Margulis’ super-rigidity [M91, Section VII.5] (see also [BoW00, Section XIII.4.6]), this is the most interesting case, when the real rank of the locally symmetric space is \(\geqslant 2\).

  3. If G is semisimple or more generally if G has a compact centre, then all the representations of G has an admissible metric ([MatMu63, Lemma 3.1], Proposition 2.9).

  4. More precisely, we need assume that the Casimir of \(\mathfrak {g}\) acts on \(\rho \) as a scalar.

  5. By [BSh19, Theorem 2.3], this is indeed independent of the choice of \(g_{\gamma }\).

  6. The quantity \(\ell _{[\gamma ]}\) defined in (2.16) depends only on the conjugacy class of \(\gamma \) in G. So they are well defined on the conjugacy classes of \(\Gamma \).

  7. A more general construction for the Selberg zeta function is given in [Sh20], which is associated to \(\eta \) and to an arbitrary representation of \(\rho :\Gamma \rightarrow {{\,\mathrm{\mathrm GL}\,}}_{r}(\mathbf {C})\).

  8. They are called Dirac cohomology of E (see [HuPa06]).


  1. Bénard, L., Dubois, J., Heusener, M., Porti, J.: Asymptotics of twisted Alexander polynomials and hyperbolic volume. Indiana Univ. Math. J. arXiv:1912.12946 (2019)

  2. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, Grundlehren Text Editions. Springer, Berlin (2004). Corrected reprint of the 1992 original

  3. Bismut, J.-M.: Hypoelliptic Laplacian and Orbital Integrals, Annals of Mathematics Studies, vol. 177. Princeton University Press, Princeton (2011)

    Book  Google Scholar 

  4. Bismut, J.-M., Ma, X., Zhang, W.: Opérateurs de Toeplitz et torsion analytique asymptotique. C. R. Math. Acad. Sci. Paris 349(17–18), 977–981 (2011)

    Article  MathSciNet  Google Scholar 

  5. Bismut, J.-M., Ma, X., Zhang, W.: Asymptotic torsion and Toeplitz operators. J. Inst. Math. Jussieu 16(2), 223–349 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bismut, J.-M., Shen, S.: Geometric orbital integrals and the center of the enveloping algebra. arXiv:1910.11731 (2019)

  7. Bismut, J.-M., Zhang, W.: An extension of a theorem by Cheeger and Müller, Astérisque (1992), no. 205, 235, With an appendix by François Laudenbach

  8. Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups, second ed., Mathematical Surveys and Monographs, vol. 67. American Mathematical Society, Providence (2000)

  9. Borns-Weil, Y., Shen, S.: Dynamical zeta functions in the nonorientable case. arXiv:2007.08043 (2020)

  10. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups, Graduate Texts in Mathematics, vol. 98. Springer, New York (1985)

  11. Bröcker, U.: Die Ruellesche Zetafunktion für \(G\)-induzierte Anosov-Flüsse. Ph.D. thesis, Humboldt-Universität Berlin, Berlin (1998)

  12. Cappell, S.E., Miller, E.Y.: Complex-valued analytic torsion for flat bundles and for holomorphic bundles with \((1,1)\) connections. Commun. Pure Appl. Math. 63(2), 133–202 (2010)

    Article  MathSciNet  Google Scholar 

  13. Cekić, M., Dyatlov, S., Küster, B., Paternain, G.P.: The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds. arXiv:2009.08558 (2020)

  14. Cheeger, J.: Analytic torsion and the heat equation. Ann. Math. (2) 109(2), 259–322 (1979)

    Article  MathSciNet  Google Scholar 

  15. Dai, X., Yu, J.: Comparison between two analytic torsions on orbifolds. Math. Z. 285(3–4), 1269–1282 (2017)

    Article  MathSciNet  Google Scholar 

  16. Dang, N.V., Guillarmou, C., Rivière, G., Shen, S.: The Fried conjecture in small dimensions. Invent. Math. 220(2), 525–579 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  17. de Rham, G.: Complexes à automorphismes et homéomorphie différentiable. Ann. Inst. Fourier Grenoble 2(1950), 51–67 (1951)

    MATH  Google Scholar 

  18. Duistermaat, J.J., Kolk, J.A.C., Varadarajan, V.S.: Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52(1), 27–93 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  19. Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. (4) 49(3), 543–577 (2016)

    Article  MathSciNet  Google Scholar 

  20. Dyatlov, S., Zworski, M.: Ruelle zeta function at zero for surfaces. Invent. Math. 210(1), 211–229 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  21. Franz, W.: Über die Torsion einer Überdeckung. J. Reine Angew. Math. 173, 245–254 (1935). German

    MathSciNet  MATH  Google Scholar 

  22. Fried, D.: Analytic torsion and closed geodesics on hyperbolic manifolds. Invent. Math. 84(3), 523–540 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  23. Fried, D.: Fuchsian groups and Reidemeister torsion, The Selberg trace formula and related topics (Brunswick, Maine, 1984), Contemp. Math., vol. 53, pp. 141–163. American Mathematical Society, Providence (1986)

  24. Fried, D.: Lefschetz formulas for flows, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, pp. 19–69. American Mathematical Society, Providence (1986)

  25. Gel’fand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont. (1969)

  26. Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Ann. Math. (2) 178(2), 687–773 (2013)

    Article  MathSciNet  Google Scholar 

  27. Guruprasad, K., Haefliger, A.: Closed geodesics on orbifolds. Topology 45(3), 611–641 (2006)

    Article  MathSciNet  Google Scholar 

  28. Hecht, H., Schmid, W.: Characters, asymptotics and \(n\)-homology of Harish-Chandra modules. Acta Math. 151(1–2), 49–151 (1983)

    Article  MathSciNet  Google Scholar 

  29. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  Google Scholar 

  30. Huang, J.-S., Pandžić, P.: Dirac Operators in Representation Theory, Mathematics: Theory & Applications. Birkhäuser Boston Inc, Boston (2006)

    MATH  Google Scholar 

  31. Knapp, A.W.: Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36. Princeton University Press, Princeton, NJ (1986). An overview based on examples

  32. Knapp, A.W.: Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140. Birkhäuser Boston, Inc., Boston (2002)

  33. Knapp, A.W., Vogan Jr., D.A.: Cohomological Induction and Unitary Representations, Princeton Mathematical Series, vol. 45. Princeton University Press, Princeton (1995)

    Book  Google Scholar 

  34. Kostant, B.: On Macdonald’s \(\eta \)-function formula, the Laplacian and generalized exponents. Adv. Math. 20(2), 179–212 (1976)

    Article  MathSciNet  Google Scholar 

  35. Kostant, B.: Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the \(\rho \)-decomposition \(C(\mathfrak{g})={\rm End}\, V_\rho \otimes C(P)\), and the \(\mathfrak{g}\)-module structure of \(\bigwedge \mathfrak{g}\). Adv. Math. 125(2), 275–350 (1997)

    Article  MathSciNet  Google Scholar 

  36. Ma, X.: Orbifolds and analytic torsions. Trans. Am. Math. Soc. 357(6), 2205–2233 (2005). electronic

    Article  MathSciNet  Google Scholar 

  37. Ma, X., Geometric hypoelliptic Laplacian and orbital integrals [after Bismut, Lebeau and Shen], no. 407. Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120–1135. Exp. No. 1130, 333–389 (2019)

  38. Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Springer, Berlin (1991)

    Google Scholar 

  39. Matsushima, Y.: A formula for the Betti numbers of compact locally symmetric Riemannian manifolds. J. Differ. Geom. 1, 99–109 (1967)

    Article  MathSciNet  Google Scholar 

  40. Matsushima, Y., Murakami, S.: On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds. Ann. Math. (2) 78, 365–416 (1963)

    Article  MathSciNet  Google Scholar 

  41. Milnor, J.: Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), pp. 115–133. Prindle, Weber & Schmidt, Boston (1968)

  42. Moscovici, H., Stanton, R.J.: \(R\)-torsion and zeta functions for locally symmetric manifolds. Invent. Math. 105(1), 185–216 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  43. Müller, W.: Analytic torsion and \(R\)-torsion of Riemannian manifolds. Adv. Math. 28(3), 233–305 (1978)

    Article  MathSciNet  Google Scholar 

  44. Müller, W.: Analytic torsion and \(R\)-torsion for unimodular representations. J. Am. Math. Soc. 6(3), 721–753 (1993)

    Article  MathSciNet  Google Scholar 

  45. Müller, W.: The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry. Progr. Math., vol. 297, pp. 317–352. Birkhäuser/Springer, Basel (2012)

  46. Müller, W.: On Fried’s conjecture for compact hyperbolic manifolds. arXiv:2005.01450 (2020)

  47. Quillen, D.: Superconnections and the Chern character. Topology 24(1), 89–95 (1985)

    Article  MathSciNet  Google Scholar 

  48. Ray, D.B., Singer, I.M.: \(R\)-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)

    Article  MathSciNet  Google Scholar 

  49. Reidemeister, K.: Homotopieringe und Linsenräume. Abh. Math. Sem. Univ. Hamburg 11(1), 102–109 (1935)

    Article  MathSciNet  Google Scholar 

  50. Salamanca-Riba, S.A.: On the unitary dual of real reductive Lie groups and the \(A_{\mathfrak{q}}(\lambda )\) modules: the strongly regular case. Duke Math. J. 96(3), 521–546 (1999)

    Article  MathSciNet  Google Scholar 

  51. Satake, I.: The Gauss-Bonnet theorem for \(V\)-manifolds. J. Math. Soc. Jpn. 9, 464–492 (1957)

    Article  MathSciNet  Google Scholar 

  52. Seeley, R.T.: Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307. American Mathematical Society, Providence (1967)

  53. Selberg, A.: On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), pp. 147–164. Tata Institute of Fundamental Research, Bombay (1960)

  54. Shen, S.: Analytic torsion, dynamical zeta functions, and the Fried conjecture. Anal. PDE 11(1), 1–74 (2018)

    Article  MathSciNet  Google Scholar 

  55. Shen, S.: Analytic torsion and dynamical flow: A survey on the Fried conjecture

  56. Shen, S.: Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces. arXiv:2009.03427 (2020)

  57. Shen, S., Yu, J.: Flat vector bundles and analytic torsion on orbifolds. Commun. Anal. Geom. arXiv:1704.08369 (2017)

  58. Shen, S., Yu, J.: Morse-Smale flow, Milnor metric, and dynamical zeta function. J. Éc. polytech. Math. 8, 585–607 (2021)

    Article  MathSciNet  Google Scholar 

  59. Spilioti, P.: Selberg and Ruelle zeta functions for non-unitary twists. Ann. Global Anal. Geom. 53(2), 151–203 (2018)

    Article  MathSciNet  Google Scholar 

  60. Spilioti, P.: Functional equations of Selberg and Ruelle zeta functions for non-unitary twists. Ann. Global Anal. Geom. 58(1), 35–77 (2020)

    Article  MathSciNet  Google Scholar 

  61. Spilioti, P.: Twisted Ruelle zeta function and complex-valued analytic torsion. arXiv:2004.13474 (2020)

  62. Vogan Jr., D.A.: Unitarizability of certain series of representations. Ann. Math. (2) 120(1), 141–187 (1984)

    Article  MathSciNet  Google Scholar 

  63. Vogan Jr., D.A., Zuckerman, G.J.: Unitary representations with nonzero cohomology. Compositio Math. 53(1), 51–90 (1984)

    MathSciNet  MATH  Google Scholar 

  64. Voros, A.: Spectral functions, special functions and the Selberg zeta function. Commun. Math. Phys. 110(3), 439–465 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  65. Wotzke, A.: Die Ruellesche Zetafunktion und die analytische Torsion hyperbolischer Mannigfaltigkeiten. Ph.D. thesis, Bonn, Bonner Mathematische Schriften (2008)

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Correspondence to Shu Shen.

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Communicated by S. Dyatlov

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The author acknowledges the partial support by the Grant ANR-20-CE40-0017.

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Shen, S. Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists. Commun. Math. Phys. 387, 1215–1255 (2021).

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