Geometric & Functional Analysis GAFA

, Volume 9, Issue 3, pp 518–567 | Cite as

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

  • W. Lück
  • T. Schick

Abstract.

Suppose \( \bar{M} \) 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 \( \bar{M} \) and the \( L^2 \)-analytic torsion of the Riemannian manifold M are equal. In particular, the \( L^2 \)-topological torsion of \( \bar{M} \) 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 [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, 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.

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • W. Lück
    • 1
  • T. Schick
    • 1
  1. 1.Fachbereich Mathematik und Informatik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany, e-mail: wolfgang.lueck@math.uni-muenster.de; thomas.schick@math.uni-muenster.deDE

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