Reidemeister torsion form on character varieties

TL;DR

This paper introduces a differential form called the adjoint Reidemeister torsion on the character variety of 3-manifolds, analyzing its properties, singularities, and relation to topological features of associated surfaces.

Contribution

It defines the adjoint Reidemeister torsion as a regular volume form on the character variety and explores its vanishing behavior at singular points, linking it to topological invariants.

Findings

Torsion form is a regular volume form on the character variety.

Vanishing of torsion occurs only at singular points, with specific conditions.

Bound on vanishing order relates to the Euler characteristic of associated surfaces.

Abstract

In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion does not vanish under a generic hypothesis on the Alexander polynomial, else we relate the vanishing order with the type of singularity. Finally we consider the ideal points added after compactification of the character variety. We bound the vanishing order of the torsion by the Euler characteristic of an essential surface associated to the ideal point by the Culler-Shalen theory. As a corollary we obtain an unexpected relation between the topology of those surfaces and the topology of the character variety.