Title: 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 Tn(M; η)}. Roughly speaking, Tn(M; η) is the Reidemeister torsion of M with re...For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants Tn(M; η)}. Roughly speaking, Tn(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 | Tn(M; η)| ∼ −n2 (Vol M/4π), extending thus the result obtained by Müller for the compact case. Concerning the latter, we prove that the sequence {| Tn(M; η)|} determines the complex-length spectrum of the manifold up to complex conjugation.Read More