#### Volume 20, issue 6 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Reidemeister torsion form on character varieties

### Léo Bénard

Algebraic & Geometric Topology 20 (2020) 2821–2884
DOI: 10.2140/agt.2020.20.2821
##### Abstract

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 rational volume form. Then we show that the torsion form has poles only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion has no pole under a generic hypothesis on the Alexander polynomial; otherwise, we relate the order of the pole 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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/agt

We have not been able to recognize your IP address 34.239.167.149 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.

##### Keywords
Reidemeister torsion, character varieties, $3$–manifolds, Culler–Shalen theory
Primary: 57M25
Secondary: 57M27