L2-torsion of hyperbolic manifolds of finite volume

Abstract

Suppose M̄ 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 L2-topological torsion of M̄ and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M̄ 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 L2-topological torsion of compact L2-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.