Reidemeister torsion form on character varieties

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.