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

TL;DR

This paper establishes a connection between the degree of the 2-torsion function of 3-manifolds and the Thurston norm, linking algebraic invariants to geometric topology.

Contribution

It proves that the degree of the 2-torsion function equals the Thurston norm for certain 3-manifolds, revealing a new relationship between torsion invariants and topology.

Findings

Degree of the 2-torsion function matches the Thurston norm.

The 2-torsion function's asymptotic behavior encodes geometric information.

Extension of previous work relating torsion and volume.

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.