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

Abstract

Suppose M 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 L 2 -topological torsion of M and the L 2 -analytic torsion of the Riemannian manifold M are equal. In particular, the L 2 -topological torsion of M 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 [14]. In dimension 3 this proves the conjecture [18, Conjecture 2.3] or [16, conjecture 7.7] which gives a complete calculation of the L 2 -topological torsion of compact L 2 -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.Key words: L 2 -torsion, hyperbolic manifolds, 3-manifolds, manifolds with boundary AMS-classification number: 58G11 IntroductionIn this paper we study L 2 -analytic torsion of a compact connected manifold M with boundary such that the interior M comes with a complete hyperbolic metric of finite volume.Notation 0.1. Let m be the dimension of M. From hyperbolic geometry we know [1, Chapter D3] that M can be written aswhere M 0 is a compact manifold with boundary and E 0 is a finite disjoint union of hyperbolic ends [0, ∞) × F j . Here each F j is a closed flat manifold and the metric on the end is the warped product du 2 + e −2u dx 2 1

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