In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.

Abstract. In this paper we study the asymptotic behavior of the analytic torsion for compact oriented hyperbolic manifolds with respect to certain rays of irreducible representations.

In this paper we define a regularized version of the analytic torsion for arithmetic quotients of a symmetric space of non-positive curvature. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator.

Abstract. In this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds. We consider sequences of coverings X i of a fixed hyperbolic orbifold X 0 . Our main result is that for certain sequences of coverings and strongly acyclic flat bundles, the analytic torsion divided by the index of the covering, converges to the L 2 -torsion. Our results apply to certain sequences of arithmetic groups, in particular to sequences of principal congruence subgroups of SO 0 (d, 1)(Z) and to sequences of principal congruence subgroups or Hecke subgroups of Bianchi groups.

We study Selberg zeta functions Z(s, σ) 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 halfspace 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.