Selberg and Ruelle zeta functions and the relative analytic torsion on complete odd-dimensional hyperbolic manifolds of finite volume

Abstract

Let X be a complete hyperbolic manifold of finite volume and of odd dimension d. Firstly, we study Selberg zeta functions on X, prove that these functions have a meromorphic continuation to the entire complex plane and describe their singularities. Secondly, we define the relative or regularized analytic torsion of X associated to certain representations of its fundamental group. We investigate the asymptotic behaviour of this torsion with respect to special sequences of representations. Finally, if X is 3-dimensional, we establish a relation between the regularized analytic torsion and the behaviour of a twisted Ruelle zeta function at 0. Our work generalizes results of Fried, Bunke and Olbrich, Bröcker and Wotzke to the non-compact case and results of Müller to the non-compact and higher-dimensional situation.