Selberg zeta function and relative analytic torsion for hyperbolic odd-dimensional orbifolds

Abstract

In this thesis we study the Selberg zeta functions and the analytic torsion of hyperbolic odd-dimensional orbifolds $Gamma backslash mathbb{H}^{2n+1}$. In the first part of the thesis we restrict ourselves to compact orbifolds and establish a version of the Selberg trace formula for non-unitary representations of $Gamma$. We study Selberg zeta functions on $Gamma backslash mathbb{H}^{2n+1}$, prove that these functions admit a meromorphic continuation to $C$ and describe their singularities. In the second part we define the analytic torsion of a compact orbifold $Gamma backslash mathbb{H}^{2n+1}$ associated to the restriction of a certain representation of $G$ to $Gamma$. Further we investigate the asymptotic behavior of this torsion with respect to special sequences of representations of $G$. In the third part we extend the results of the second part to hyperbolic odd-dimensional orbifolds of finite volume under the assumption that the orbifold is 3-dimensional.<br /> Our work generalizes the results of Mueller to compact orbifolds, results of Bunke and Olbrich to compact orbifolds and non-unitary representations of $Gamma$, and results of Mueller and Pfaff to compact and finite-volume 3-dimensional orbifolds.