Higher‐dimensional Reidemeister torsion invariants for cusped hyperbolic 3‐manifolds. (4th July 2013) - Menal-Ferrer, Pere; Porti, Joan
Abstract : 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 Müller for the compact case. Concerning the latter, we prove that the sequence {| T n ( M ; η)|} determines the complex‐length spectrum of the manifold up to complex conjugation.