The divisor of Selberg's zeta function for Kleinian groups

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\\sp n+1 together with the Euler characteristic of...

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Bibliographic Details
Published in:Duke mathematical journal 2001-02, Vol.106 (2), p.321-390
Main Authors: Patterson, S. J., Perry, Peter A.
Format: Article
Language:English
Subjects:
(Ruelle-Frobenius) transfer operators
11F72
11M36
22E40
32Nxx
37C30
37D35
and other functional analytic techniques in dynamical systems
applications to spectral theory
Dirichlet series
Discrete subgroups of Lie groups [See also 20Hxx
Eisenstein series
equilibrium states
etc. Explicit formulas
Selberg trace formula
Selberg zeta functions and regularized determinants
Spectral theory
Thermodynamic formalism
variational principles
Zeta functions
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Summary: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\\sp 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.
ISSN:0012-7094
1547-7398
DOI:10.1215/s0012-7094-01-10624-8