Duke Mathematical Journal

The divisor of Selberg's zeta function for Kleinian groups

S. J. Patterson and Peter A. Perry

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We compute the divisor of Selberg's zeta function for convex cocompact, torsion-free discrete groups Γ acting on a real hyperbolic space of dimension n+1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on $X = Γ \backslash \mathbb{H}^{n+1}$ together with the Euler characteristic of X compactified to a manifold with boundary. If n is even, the singularities of the zeta function associated to the Euler characteristic of X are identified using work of U. Bunke and M. Olbrich.

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Duke Math. J., Volume 106, Number 2 (2001), 321-390.

First available in Project Euclid: 13 August 2004

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Zentralblatt MATH identifier

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37D35: Thermodynamic formalism, variational principles, equilibrium states


Patterson, S. J.; Perry, Peter A. The divisor of Selberg's zeta function for Kleinian groups. Duke Math. J. 106 (2001), no. 2, 321--390. doi:10.1215/S0012-7094-01-10624-8. https://projecteuclid.org/euclid.dmj/1092403918

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