Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume

Abstract

Abstract We study Selberg zeta functions Z(s,σ) associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete oriented odd-dimensional hyperbolic spin manifold of finite volume. We assume a certain condition on the fundamental group of the manifold. A priori, the Selberg zeta functions are defined only for s in some right half-plane of ℂ. We will prove that for any locally homogeneous bundle the functions Z(s,σ) have a meromorphic continuation to ℂ and we will give a complete description of their singularities in terms of spectral data of the underlying manifold. Our work generalizes results of Bunke and Olbrich to the non-compact situation. As an application of our results one can compare the normalized Reidemeister torsions on hyperbolic 3-manifolds with cusps which were introduced by Menal-Ferrer and Porti to the corresponding regularized analytic torsions.