On the positivity of twisted $L^2$-torsion for 3-manifolds

TL;DR

This paper proves that for certain 3-manifolds with infinite fundamental groups, the twisted $L^2$-torsion function is strictly positive and continuous on specific representation subvarieties, advancing understanding of torsion invariants.

Contribution

It establishes the positivity and continuity of the twisted $L^2$-torsion function for a class of 3-manifolds, linking geometric properties to algebraic invariants.

Findings

Twisted $L^2$-torsion is non-negative for these manifolds.

Strict positivity of the torsion function when the fundamental group is infinite.

Continuity of the torsion function on upper triangular representation subvarieties.

Abstract

For any compact orientable irreducible 3-manifold $N$ with empty or incompressible toral boundary, the twisted $L^2$-torsion is a non-negative function defined on the representation variety $\operatorname{Hom}(\pi_1(N),\operatorname{SL}(n,\mathbb C))$. The paper shows that if $N$ has infinite fundamental group, then the $L^2$-torsion function is strictly positive. Moreover, this torsion function is continuous when restricted to the subvariety of upper triangular representations.