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 which is not $S^1 \times D^2$. Consider any element $\phi$ in the first cohomology of $M$ with integer 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 the degree of the $L^2$-torsion function, which is a number extracted from its asymptotic behavior at 0 and at $\infty$, agrees with the Thurston norm of $\phi$.