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A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS

Part of: Global differential geometry Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  01 July 2015

Jonathan Pfaff
Jonathan Pfaff*
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Alle 60, D-53115 Bonn, Germany (pfaff@math.uni-bonn.de)
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Abstract

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems, we define the Reidemeister torsion of the Borel–Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula, and here our approach is heavily inspired by a recent paper of Lesch.

Keywords

analytic torsionlocally symmetric manifoldsReidemeister torsion

MSC classification

Primary: 58J52: Determinants and determinant bundles, analytic torsion
Secondary: 53C35: Symmetric spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 4 , September 2017 , pp. 673 - 743
Copyright
© Cambridge University Press 2015 

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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
Berger, M., Gauduchon, P. and Mazet, E., Le spectre d’une varieté Riemannienne, Lecture Notes in Mathematics, Volume 194 (Springer, Berlin, Heidelberg, New York, 1971).Google Scholar
Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren Text Editions (Springer, Berlin, 2004). Corrected reprint of the 1992 original.Google Scholar
J. Bismut, W. Zhang (with an appendix by François Laudenbach), An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992).Google Scholar
Bismut, J. M., Ma, X. and Zhang, W., Asymptotic torsion and Toeplitz operators, preprint, 2011, available at http://www.math.jussieu.fr/∼ma/mypubli/BismutMaZhangglob.pdf.Google Scholar
Borel, A., Introduction aux groupes arithmétiques, in Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, Volume 1341 (Hermann, Paris, 1969).Google Scholar
Borel, A., Regularization theorems in Lie algebra cohomology. Applications, Duke Math. J. 50(3) (1983), 605–623.Google Scholar
Borel, A. and Casselman, W., L2-cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50(3) (1983), 625–647.Google Scholar
Borel, A. and Garland, H., Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105(2) (1983), 309–335.Google Scholar
Borel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (Princeton University Press, Princeton, 1980).Google Scholar
Brislawn, C., Traceable integral kernels on countably generated measure spaces, Pacific J. Math. 150(2) (1991), 229–240.Google Scholar
Brüning, J. and Lesch, M., Hilbert complexes, J. Funct. Anal. 108(1) (1992), 88–132.Google Scholar
Brüning, J. and Ma, X., An anomaly formula for Ray-Singer metrics on manifolds with boundary, Geom. Funct. Anal. 16(4) (2006), 767–837.Google Scholar
Brüning, J. and Ma, X., On the gluing formula for the analytic torsion, Math. Z. 273(3–4) (2013), 1085–1117.Google Scholar
Calegari, F. and Venkatesh, A., A torsion Jacquet–Langlands correspondence, preprint, 2012, arXiv:1212.3847.Google Scholar
Carslaw, H. and Jaeger, J., Conduction of Heat in Solids, 2nd ed. (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988). Reprint.Google Scholar
Cheeger, J., Analytic torsion and the heat equation, Ann. of Math. (2) 109(2) (1979), 259–322.Google Scholar
Cheeger, J., Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18(4) (1983), 575–657.Google Scholar
Chernoff, P., Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401–414.Google Scholar
Donnelly, H., Spectral geometry for certain noncompact Riemannian manifolds, Math. Z. 169(1) (1979), 63–76.Google Scholar
Donnelly, H., Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23(3) (1979), 485–496.CrossRefGoogle Scholar
van Est, W., A generalization of the Cartan-Leray spectral sequence. I, II, Indag. Math. (N.S.) 20 (1958), 399–413.Google Scholar
Gilkey, P., Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., Studies in Advanced Mathematics, (CRC Press, Boca Raton, FL, 1995).Google Scholar
Goodman, R. and Wallach, N., Representations and invariants of the classical groups, in Encyclopedia of Mathematics and its Applications vol 68 (Cambridge University Press, Cambridge, 1998).Google Scholar
Greiner, P., An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163–218.CrossRefGoogle Scholar
Harder, G., On the cohomology of discrete arithmetically defined groups, in Discrete Subgroups of Lie Groups and Applications to Moduli (Internat. Colloq., Bombay, 1973), pp. 129–160 (Oxford University Press, Bombay, 1975).Google Scholar
Harish-Chandra, Automorphic Forms on Semisimple Lie Gropus, Lecture Notes in Mathematics, Volume 62(Springer, Berlin, 1968).Google Scholar
Hassell, A., Analytic surgery and analytic torsion, Comm. Anal. Geom. 6(2) (1998), 255–289.Google Scholar
Knapp, A., Lie Groups Beyond an Introduction, 2nd ed. (Birkhäuser, Boston, 2002).Google Scholar
Knapp, A. and Stein, E., Intertwining operators for semisimple Lie groups, Ann. of Math. (2) 93 (1971), 489–578.Google Scholar
Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329–378.Google Scholar
Langlands, R., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, Volume 544 (Springer, Berlin, 1976).Google Scholar
Lesch, M., A gluing formula for the analytic torsion on singular spaces, Anal. PDE 6(1) (2013), 221–256.Google Scholar
Lott, J. and Rothenberg, M., Analytic torsion for group actions, J. Differential Geom. 34(2) (1991), 431–481.Google Scholar
Lück, W., Analytic and topological torsion for manifolds with boundary and symmetry, J. Differential Geom. 37(2) (1993), 263–322.Google Scholar
Lück, W. and Schick, T., L2-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9(3) (1999), 518–567.Google Scholar
Marshall, S. and Müller, W., On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. 162(5) (2013), 863–888.Google Scholar
Melrose, R. B., The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, Volume 4 (A K Peters, Ltd., Wellesley, MA, 1993).Google Scholar
Menal-Ferrer, P. and Porti, J., Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol., in press, doi:10.1112/jtopol/jtt024.Google Scholar
Miatello, R. J., The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature, Trans. Amer. Math. Soc. 260(1) (1980), 1–33.Google Scholar
Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.Google Scholar
Moscovici, H. and Stanton, R., R-torsion and zeta functions for locally symmetric manifolds, Inv. Math. 105 (1991), 185–216.Google Scholar
Müller, W., Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math. 28(3) (1978), 233–305.Google Scholar
Müller, W., Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197–288.CrossRefGoogle Scholar
Müller, W., Manifolds with Cusps of Rank One. Spectral Theory and L 2 -index Theorem, Lecture Notes in Mathematics, Volume 1244 (Springer, Berlin, 1987).Google Scholar
Müller, W., Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721–753.Google Scholar
Müller, W., Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys. 192(2) (1998), 309–347.Google Scholar
Müller, W. and Pfaff, J., Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. 263(9) (2012), 2615–2675.Google Scholar
Müller, W. and Pfaff, J., On the asymptotics of the Ray-Singer analytic torsion for compact hyperbolic manifolds, Int. Math. Res. Not. IMRN (13) (2013), 2945–2984.Google Scholar
Müller, W. and Pfaff, J., Analytic torsion and L2-torsion of compact locally symmetric manifolds, J. Differential Geom. 95(1) (2013), 71–119.Google Scholar
Müller, W. and Pfaff, J., On the growth of torsion in the cohomology of arithmetic groups, Math. Ann. 359(1–2) (2014), 537–555.Google Scholar
Müller, W. and Pfaff, J., The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds, J. Funct. Anal. 267(8) (2014), 2731–2786.Google Scholar
Müller, W. and Salomonsen, G., Scattering theory for the Laplacian on manifolds with bounded curvature, J. Funct. Anal. 253(1) (2007), 158–206.Google Scholar
Murakami, M., On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365–416.Google Scholar
Park, J., Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps, J. Funct. Anal. 257(6) (2009), 1713–1758.Google Scholar
Paquet, L., Problemes mixtes pour le systme de Maxwell, Ann. Fac. Sci. Toulouse Math. (5) 4(2) (1982), 103–141.Google Scholar
Pfaff, J., Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume, J. Reine Angew. Math. 703 (2015), 115–147.Google Scholar
Pfaff, J., Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z., to appear, preprint, 2012, arXiv:1206.0228.Google Scholar
Raimbault, J., Asymptotics of analytic torsion for hyperbolic three-manifolds, preprint, 2012, arXiv:1212.3161.Google Scholar
Ray, D. and Singer, I. M., R-torsion and the Laplacian on Riemannian manifolds., Adv. Math. 7 (1971), 145–210.Google Scholar
Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, Volume 988 (Springer, Berlin, 1983).Google Scholar
Selberg, A., Harmonic analysis, in Collected Papers vol. I, pp. 626–674 (Springer, Berlin, 1989).Google Scholar
Taylor, M., Pseudo-differential Operators (Princeton University Press, Princeton, NJ, 1981).Google Scholar
Vertman, B., Analytic torsion of a bounded generalized cone, Comm. Math. Phys. 290(3) (2009), 813–860.CrossRefGoogle Scholar
Vishik, S., Generalized Ray–Singer conjecture. I. A manifold with a smooth boundary, Comm. Math. Phys. 167(1) (1995), 1–102.Google Scholar
Warner, G., Selberg’s trace formula for nonuniform lattices: the R-rank one case, in Studies in algebra and number theory, Adv. Math. Suppl. Stud., Volume 6 (Academic Press, New York-London, 1979).Google Scholar
Warner, G., Harmonic analysis on semi-simple Lie groups, I, in Die Grundlehren der mathematischen Wissenschaften, Band Volume 188 (Springer, New York–Heidelberg, 1972).Google Scholar
Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics, Volume 61 (Springer, New York, Berlin, 1978).Google Scholar