$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume

Suppose is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold M are equal. In particular, the -topological torsion of is proportional to the hyperbolic 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]. In dimension 3 this proves the conjecture Lü2, Conjecture 2.3] or LLü, Conjecture 7.7] which gives a complete calculation of the -topological torsion of compact -acyclic 3-manifolds which admit a geometric JSJT-decomposition.?In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Submitted: March 1998, revised: July 1998.