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 dimension 3, 5 and 7. In dimension 3 this proves the conjecture Of Lott and Lueck which gives a complete calculation of the
L
2
-topological torsion of compact
L
2
-acyclic 3-manifolds which admit a geometric torus-decomposition. In an appendix we give a counterexample to an extension of the Cheeger-Mueller 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.
Keywords: L^2-torsion, hyperbolic manifolds, 3-manifolds