The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein
type: Item
instance of: scholarly article
title: The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein
zbMATH Open document ID: 1012.11083
DOI: 10.1215/S0012-7094-01-10624-8
author: Peter Perry
author: Samuel James Patterson
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Selberg's zeta-function is defined in terms of the length spectrum of a compact or, more generally, a finite volume hyperbolic surface. Its divisor is determined by the eigenvalues and scattering poles of the Laplacian and the Euler characteristic of the surface. The authors note ``this relationship may be regarded as one form of Selberg's trace formula for a compact or finite volume surface and as a particularly precise quantization of an ergodic system, namely, the geodesic flow''. The authors extend this relationship to convex cocompact hyperbolic manifolds in all dimensions. They show the divisor of the Selberg zeta-function is determined by the eigenvalues and scattering poles of the Laplacian together with the Euler characteristic of the corresponding compactified manifold with boundary. In even dimensions, this uses work of Bunke and Olbrich. The contents of the paper are as follows: \S 1 Introduction. \S 2 Symbolic dynamics and Selberg's zeta-function. \S 3 Geometry at infinity (coordinate neighborhoods, envelopes of horospheres, normal flow). \S 4 Review of scattering theory (model space, resolvant, eigenfunctions, operator). \S 5 The `logarithmic derivative' of the scattering operator.\S 6 The logarithmic derivative of the zeta-function. \S 7 Computation of the divisor (spectral term, topological term, residues in odd and even dimensions). Appendix A: An asymptotic volume formula for convex cocompact hyperbolic manifolds (by C. Epstein). Appendix B: The scattering operator and zeta-function for a class of cylindrical manifolds.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 1820780
zbMATH Keywords: geodesic flow
zbMATH Keywords: convex cocompact hyperbolic manifolds
zbMATH Keywords: divisor
zbMATH Keywords: Selberg zeta-function
zbMATH Keywords: eigenvalues
zbMATH Keywords: scattering poles
zbMATH Keywords: Laplacian
zbMATH Keywords: Euler characteristic
zbMATH Keywords: symbolic dynamics
MaRDI profile type: Publication
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The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein (full work available at URL)
full work available at URL: s0012-7094-01-10624-8

The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein (OpenAlex ID)
OpenAlex ID: W2090111652