Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group

Friedman, Joshua

doi:10.1090/S0002-9939-08-09516-6

Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group
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by Joshua S. Friedman PDF

Proc. Amer. Math. Soc. 136 (2008), 3395-3404 Request permission

Abstract:

For Kleinian groups acting on a hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups to Kleinian groups, and we give a proof that is simple and extendable to more general groups.

References

J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315, DOI 10.1007/978-3-662-03626-6
Joshua S. Friedman, The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations, Math. Z. 250 (2005), no. 4, 939–965. MR 2180383, DOI 10.1007/s00209-005-0806-9
—, The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations. Ph.D. Thesis, Stony Brook University, 2005, http://arxiv.org/abs/math.NT/0612807.
Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,R)$. Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR 0439755
Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,\,\textbf {R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197, DOI 10.1007/BFb0061302
A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321
Walter Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Ann. 167 (1966), no. 4, 292–337 (German). MR 1513277, DOI 10.1007/BF01364540
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172 (Russian). MR 665585
A. B. Venkov and P. G. Zograf, Analogues of Artin’s factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1150–1158, 1343 (Russian). MR 682487