L-2-torsion of hyperbolic manifolds of finite volume

Suppose (M) over bar is a compact connected odd-dimensional manifold with b oundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L-2-topological torsion of (M) over bar and the L-2-analytic torsion of the Riemannian manifold M are equal. In particu lar, the L-2-topological torsion of (M) over bar is proportional to the hyp erbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. I n dimension 3 this proves the conjecture [Lu2, Conjecture 2.3] or [LLu, Con jecture 7.7] which gives a complete calculation of the L-2-topological tors ion of compact L-2-acyclic 3-manifolds which admit a geometric JSJT-decompo sition. In an appendix we give a counterexample to an extension of the Cheeger-Mull er theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, ev en if the Euler characteristic of the boundary vanishes.