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  • Jonathan Pfaff (a1)

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.

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1. Bergeron, N. and Venkatesh, A., The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12(2) (2013), 391447.
2. 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).
3. Berline, N., Getzler, E. and Vergne, M., Heat Kernels and Dirac Operators, Grundlehren Text Editions (Springer, Berlin, 2004). Corrected reprint of the 1992 original.
4. J. Bismut, W. Zhang (with an appendix by François Laudenbach), An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992).
5. Bismut, J. M., Ma, X. and Zhang, W., Asymptotic torsion and Toeplitz operators, preprint, 2011, available at∼ma/mypubli/BismutMaZhangglob.pdf.
6. 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).
7. Borel, A., Regularization theorems in Lie algebra cohomology. Applications, Duke Math. J. 50(3) (1983), 605623.
8. Borel, A. and Casselman, W., L2-cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50(3) (1983), 625647.
9. Borel, A. and Garland, H., Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105(2) (1983), 309335.
10. Borel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (Princeton University Press, Princeton, 1980).
11. Brislawn, C., Traceable integral kernels on countably generated measure spaces, Pacific J. Math. 150(2) (1991), 229240.
12. Brüning, J. and Lesch, M., Hilbert complexes, J. Funct. Anal. 108(1) (1992), 88132.
13. Brüning, J. and Ma, X., An anomaly formula for Ray-Singer metrics on manifolds with boundary, Geom. Funct. Anal. 16(4) (2006), 767837.
14. Brüning, J. and Ma, X., On the gluing formula for the analytic torsion, Math. Z. 273(3–4) (2013), 10851117.
15. Calegari, F. and Venkatesh, A., A torsion Jacquet–Langlands correspondence, preprint, 2012, arXiv:1212.3847.
16. 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.
17. Cheeger, J., Analytic torsion and the heat equation, Ann. of Math. (2) 109(2) (1979), 259322.
18. Cheeger, J., Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18(4) (1983), 575657.
19. Chernoff, P., Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401414.
20. Donnelly, H., Spectral geometry for certain noncompact Riemannian manifolds, Math. Z. 169(1) (1979), 6376.
21. Donnelly, H., Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23(3) (1979), 485496.
22. van Est, W., A generalization of the Cartan-Leray spectral sequence. I, II, Indag. Math. (N.S.) 20 (1958), 399413.
23. 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).
24. 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).
25. Greiner, P., An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163218.
26. 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. 129160 (Oxford University Press, Bombay, 1975).
27. Harish-Chandra, Automorphic Forms on Semisimple Lie Gropus, Lecture Notes in Mathematics, Volume 62(Springer, Berlin, 1968).
28. Hassell, A., Analytic surgery and analytic torsion, Comm. Anal. Geom. 6(2) (1998), 255289.
29. Knapp, A., Lie Groups Beyond an Introduction, 2nd ed. (Birkhäuser, Boston, 2002).
30. Knapp, A. and Stein, E., Intertwining operators for semisimple Lie groups, Ann. of Math. (2) 93 (1971), 489578.
31. Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329378.
32. Langlands, R., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, Volume 544 (Springer, Berlin, 1976).
33. Lesch, M., A gluing formula for the analytic torsion on singular spaces, Anal. PDE 6(1) (2013), 221256.
34. Lott, J. and Rothenberg, M., Analytic torsion for group actions, J. Differential Geom. 34(2) (1991), 431481.
35. Lück, W., Analytic and topological torsion for manifolds with boundary and symmetry, J. Differential Geom. 37(2) (1993), 263322.
36. Lück, W. and Schick, T., L2-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9(3) (1999), 518567.
37. Marshall, S. and Müller, W., On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. 162(5) (2013), 863888.
38. Melrose, R. B., The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, Volume 4 (A K Peters, Ltd., Wellesley, MA, 1993).
39. 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.
40. 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), 133.
41. Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358426.
42. Moscovici, H. and Stanton, R., R-torsion and zeta functions for locally symmetric manifolds, Inv. Math. 105 (1991), 185216.
43. Müller, W., Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math. 28(3) (1978), 233305.
44. Müller, W., Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197288.
45. Müller, W., Manifolds with Cusps of Rank One. Spectral Theory and L 2 -index Theorem, Lecture Notes in Mathematics, Volume 1244 (Springer, Berlin, 1987).
46. Müller, W., Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc. 6 (1993), 721753.
47. Müller, W., Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys. 192(2) (1998), 309347.
48. Müller, W. and Pfaff, J., Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. 263(9) (2012), 26152675.
49. 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), 29452984.
50. Müller, W. and Pfaff, J., Analytic torsion and L2-torsion of compact locally symmetric manifolds, J. Differential Geom. 95(1) (2013), 71119.
51. Müller, W. and Pfaff, J., On the growth of torsion in the cohomology of arithmetic groups, Math. Ann. 359(1–2) (2014), 537555.
52. Müller, W. and Pfaff, J., The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds, J. Funct. Anal. 267(8) (2014), 27312786.
53. Müller, W. and Salomonsen, G., Scattering theory for the Laplacian on manifolds with bounded curvature, J. Funct. Anal. 253(1) (2007), 158206.
54. Murakami, M., On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365416.
55. Park, J., Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps, J. Funct. Anal. 257(6) (2009), 17131758.
56. Paquet, L., Problemes mixtes pour le systme de Maxwell, Ann. Fac. Sci. Toulouse Math. (5) 4(2) (1982), 103141.
57. Pfaff, J., Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume, J. Reine Angew. Math. 703 (2015), 115147.
58. Pfaff, J., Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps, Math. Z., to appear, preprint, 2012, arXiv:1206.0228.
59. Raimbault, J., Asymptotics of analytic torsion for hyperbolic three-manifolds, preprint, 2012, arXiv:1212.3161.
60. Ray, D. and Singer, I. M., R-torsion and the Laplacian on Riemannian manifolds., Adv. Math. 7 (1971), 145210.
61. Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, Volume 988 (Springer, Berlin, 1983).
62. Selberg, A., Harmonic analysis, in Collected Papers vol. I, pp. 626674 (Springer, Berlin, 1989).
63. Taylor, M., Pseudo-differential Operators (Princeton University Press, Princeton, NJ, 1981).
64. Vertman, B., Analytic torsion of a bounded generalized cone, Comm. Math. Phys. 290(3) (2009), 813860.
65. Vishik, S., Generalized Ray–Singer conjecture. I. A manifold with a smooth boundary, Comm. Math. Phys. 167(1) (1995), 1102.
66. 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).
67. Warner, G., Harmonic analysis on semi-simple Lie groups, I, in Die Grundlehren der mathematischen Wissenschaften, Band Volume 188 (Springer, New York–Heidelberg, 1972).
68. Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics, Volume 61 (Springer, New York, Berlin, 1978).
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