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THE EQUIVARIANT CHEEGER–MÜLLER THEOREM ON LOCALLY SYMMETRIC SPACES

Part of: Homology and homotopy of topological groups and related structures Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  02 October 2014

Michael Lipnowski
Michael Lipnowski*
Affiliation:
Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320, USA (malipnow@math.duke.edu)
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Abstract

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion, and demonstrate that Bismut–Zhang’s equivariant Cheeger–Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and torsion cohomology comparison for arithmetic locally symmetric spaces to an equivariant setting.

Keywords

Reidemeister torsionanalytic torsionCheeger–Müller theoremequivariantlocally symmetric space

MSC classification

Primary: 57T99: None of the above, but in this section
Secondary: 58J52: Determinants and determinant bundles, analytic torsion
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 165 - 202
Copyright
© Cambridge University Press 2014 

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References

Bergeron, N. and Venkatesh, A., The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12(2) (2013), 391–447.Google Scholar
Bismut, J.-M. and Cheeger, J., Transgressed Euler classes of SL 2n(ℤ) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. Sci. Éc. Norm. Supér. (4) 25(4) (1992), 335–391.CrossRefGoogle Scholar
Bismut, J.-M. and Zhang, W., An extension of a theorem by Cheeger and Müller, Astérisque (205) (1992), (With an appendix by François Laudenbach).Google Scholar
Bismut, J.-M. and Zhang, W., Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal. 4(2) (1994), 136–212.Google Scholar
Calegari, F. and Venkatesh, A., A Torsion Jacquet–Langlands correspondence, Submitted; arXiv:http://arxiv.org/abs/1212.3847.Google Scholar
Cheeger, J., Analytic torsion and the heat equation, Ann. of Math. (2) 109(2) (1979), 259–322.CrossRefGoogle Scholar
Illman, S., Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Ann. 233(3) (1978), 199–220.CrossRefGoogle Scholar
Jacquet, H. and Langlands, R., Automorphic Forms on GL 2, Lecture Notes in Mathematics, Volume 114 (Springer, Berlin, 1970).CrossRefGoogle Scholar
Knudsen, F. and Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39(1) (1976), 19–55.Google Scholar
Lipnowski, M., Equivariant Torsion and Base Change, Submitted; arXiv:http://arxiv.org/abs/1312.2540.Google Scholar
Lott, J. and Rothenberg, M., Analytic torsion for group actions, J. Differential Geom. 34(2) (1991), 431–481.CrossRefGoogle Scholar
Lück, W., Analytic and topological torsion for manifolds with boundary and symmetry, J. Differential Geom. 37(2) (1993), 263–322.CrossRefGoogle Scholar
Müller, W., Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math. 28(3) (1978), 233–305.Google Scholar
Müller, W., Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc. 6(3) (1993), 721–753.Google Scholar
Nicolaescu, L., An Invitation to Morse Theory, second edition (Universitext). (Springer, New York, 2011).CrossRefGoogle Scholar
Oesterlé, J., Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math. 78(1) (1984), 13–88.Google Scholar
Ray, D. B. and Singer, I. M., R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210.Google Scholar
Sengün, M. H., On the integral cohomology of Bianchi groups, Exp. Math. 20(4) (2011), 487–505.CrossRefGoogle Scholar