For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {T n (M ; η)}. Roughly speaking, T n (M ; η) is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by η, and the n-dimensional, irreducible, complex representation of SL(2, C). In the present work, we focus on two aspects of this invariant: its asymptotic behaviour and its relationship with the complex-length spectrum of the manifold. Concerning the former, we prove that for suitable spin structures, log |T n (M ; η)| ∼ −n 2 Vol M 4π , extending thus the result obtained by W. Müller for the compact case in [Mül]. Concerning the latter, we prove that the sequence {|T n (M ; η)|} determines the complex-length spectrum of the manifold up to complex conjugation.

Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2, C) with the n-dimensional irreducible representation of SL(2, C) in SL(n, C). In this paper we give local coordinates of the SL(n, C)-character variety around this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.

For a complete hyperbolic three manifold M , we consider the representations of π 1 (M ) obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2, C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using techniques of Matsushima-Murakami. We give some applications to local rigidity.

Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift of the holonomy to SL.2; C/ is invariant under mutation along a surface of genus 2, hence also under mutation along a Conway sphere.