Given a finite dimensional irreducible complex representation of G=SOo(d,1), one can associate a canonical flat vector bundle E together with a canonical bundle metric h to any finite volume hyperbolic manifold X of dimension d. For d odd and provided X satisfies some mild hypotheses, we will explain how, by looking at a family of compact manifolds degenerating to X in a suitable sense, one can obtain a formula relating the analytic torsion of (X,E,h) with the Reidemeister torsion of an associated manifold with boundary. As an application, we will indicate how, in the arithmetic setting, this formula can be used to derive exponential growth of torsion in cohomology for various sequences of congruence subgroups. This is a joint work with Werner Mueller.