Reidemeister torsion, hyperbolic three-manifolds, and character varieties

Abstract

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional representations of $\operatorname{ SL}_2(\mathbb C)$. In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\operatorname{ PSL}_{n+1}(\mathbb C)$-characters, using that the first cohomology group with coefficients twisted by the adjoint is the tangent space to the variety of characters.