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

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element ϕin the first cohomology of M with integral coefficients. Then one can define the ϕ-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 ϕ.

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