Abstract
In this paper we consider the analytic torsion of a closed hyperbolic 3-manifold associated with the mth symmetric power of the standard representation of SL(2C) and we study its asymptotic behavior as mtends to infinity. The leading coefficient of the asymptotic formula is given by the volume of the hyperbolic 3-manifold. It follows that the Reidemeister torsion associated with the symmetric powers determines the volume of a closed hyperbolic 3-manifold.
Mathematics Subject Classification (2000). Primary: 58J52, Secondary: 11M36.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
D. Barbasch, H. Moscovici, 2-index and the trace formula, J. Funct. Analysis 53 (1983), 151–201.
R. Berndt, Representations of linear groups. An introduction based on examples from physics and number theory. Vieweg, Wiesbaden, 2007.
J.-M. Bismut, E. Vasserot, The asymptotics of the Ray-Singer analytic torsion of the symmetric powers of a positive vector bundle. Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 835–848.
A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition. Mathematical Surveys and Monographs, 67. Amer. Math. Soc., Providence, RI, 2000.
U. Bröcker, Die Ruellesche Zetafunktion fÜr -induzierte Anosov-FlÜsse, Ph.D. thesis, Humboldt-Universität Berlin, Berlin, 1998.
U. Bunke and M. Olbirch, Selberg zeta and theta functions, A differential operator approach, Akademie Verlag, Berlin, 1995.
T.A. Chapman, Topological invariance of Whitehaed torsion, Amer. J. Math. 96 (1974), 488–497.
J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. math. 84 (1986), 523–540.
D. Fried, Meromorphic zeta functions of analytic flows, Commun. Math. Phys. 174 (1995), 161–190.
A.W. Knapp, Representation theory of semisimple groups, Princeton University Press, Princeton and Oxford, 2001.
A.W. Knapp and E.M. Stein, Intertwining operators for semisimple Lie groups, Annals of Math. 93 (1971), 489–578.
B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. (2) 74 (1961) 329–387.
Matsushima, Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. 78 (1963), 365–416.
R.J. Miatello, The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature. Trans. Amer. Math. Soc. 260 (1980), 1–33.
J.J. Millson, Closed geodesics and the -invariant, Annals of Math. 108 (1978), 1–39.
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.
G.D. Mostow, Strong rigidity of locally symmetric spaces, Princeton Univ. Press and Univ. of Tokyo Press, 1973.
W. MÜller, Analytic torsion and -torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721–753.
G. Prasad, Strong rigidity of ℚ-rank 1 lattices, Invent. math. 21 (1973), 255–286.
D.B. Ray, I.M. Singer; -torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7, 145–210. (1971).
M.A. Shubin, Pseudodifferential operators and spectral theory, Second edition. Springer-Verlag, Berlin, 2001.
J. Silhan, A real analog of Kostant’s version of the Bott-Borel-Weil theorem, J. of Lie theory 14 (2004), 481–499.
A. Voros, Spectral functions, special functions and the Selberg zeta function, Commun. Math. Phys. 110 (1987), 439–465.
W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), no. 3, 357–381.
W. Thurston, Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.
N.R. Wallach, On the Selberg trace formula in the case of compact quotient, Bull. Amer. Math. Soc. 82 (1976), 171–195.
A. Wotzke, Die Ruellsche Zetafunktion und die analytische Torsion hyperbolischer Mannigfaltigkeiten, Ph.D. thesis, Bonn, 2008, Bonner Mathematische Schriften, Nr. 389.
B. Zimmermann, A note on hyperbolic 3-manifolds of the same volume, Monatsh. Math. 117, (1994), no. 1-2, 139–143.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Jeff Cheeger for his 65th birthday
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Müller, W. (2012). The Asymptotics of the Ray-Singer Analytic Torsion of Hyperbolic 3-manifolds. In: Dai, X., Rong, X. (eds) Metric and Differential Geometry. Progress in Mathematics, vol 297. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0257-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0257-4_11
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0256-7
Online ISBN: 978-3-0348-0257-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)