The L^2-torsion function and the Thurston norm of 3-manifolds

Abstract

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element \phi in the first cohomology of M with integral coefficients. Then one can define the \phi-twisted L^2-torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that its degree, which is a number extracted from its asymptotic behavior at 0 and at infinity, agrees with the Thurston norm of \phi.