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2020 Reidemeister torsion form on character varieties
Léo Bénard
Algebr. Geom. Topol. 20(6): 2821-2884 (2020). 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.

Citation

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Léo Bénard. "Reidemeister torsion form on character varieties." Algebr. Geom. Topol. 20 (6) 2821 - 2884, 2020. https://doi.org/10.2140/agt.2020.20.2821

Information

Received: 6 August 2018; Revised: 18 December 2019; Accepted: 20 January 2020; Published: 2020
First available in Project Euclid: 16 December 2020

MathSciNet: MR4185929
Digital Object Identifier: 10.2140/agt.2020.20.2821

Subjects:
Primary: 57M25
Secondary: 57M27

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 6 • 2020
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