The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein
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.
scientific article; zbMATH DE number 1820780
scientific article