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

TL;DR

This paper proves the meromorphic continuation and describes the singularities of Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume, extending previous results to non-compact cases.

Contribution

It generalizes the meromorphic continuation and spectral analysis of Selberg zeta functions to non-compact odd-dimensional hyperbolic manifolds, under certain fundamental group conditions.

Findings

Selberg zeta functions admit meromorphic continuation to all of .

Singularities are characterized by spectral data of the manifold.

Results enable comparison of Reidemeister and analytic torsions in hyperbolic 3-manifolds.

Abstract

We study Selberg zeta functions $Z(s,\sigma)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic 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-space of $\mathbb{C}$. We will prove that for any locally homogeneous bundle the functions $Z(s,\sigma)$ have a meromorphic continuation to $\mathbb{C}$ 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.