Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

Abstract

For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {τ_n(M; η)}. Roughly speaking, {τ_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 behavior and its relationship with the complex-length spectrum of the manifold. Concerning the former, we prove that for suitable spin structures, log(τ_n(M; η)) grows as -n^2 Vol(M)/4π, extending thus the result obtained by W. Mueller for the compact case. Concerning the latter, we prove that the sequence {τ_n(M; η)} determines the complex-length spectrum of the manifold up to complex conjugation.