Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

TL;DR

This paper studies higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, analyzing their asymptotic growth and their ability to determine the complex-length spectrum, extending known results to non-compact cases.

Contribution

It extends the understanding of Reidemeister torsion invariants to cusped hyperbolic 3-manifolds, establishing their asymptotic behavior and spectral determination.

Findings

log( au_n(M; \eta)) grows as -n^2 Vol(M)/4\pi for suitable spin structures

The sequence au_n(M; \eta) determines the complex-length spectrum up to conjugation

Extends Mueller's results from compact to cusped hyperbolic 3-manifolds

Abstract

For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure \eta, we consider a sequence of invariants {\tau_n(M; \eta)}. Roughly speaking, {\tau_n(M; \eta)} is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by \eta, 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(\tau_n(M; \eta)) grows as -n^2 Vol(M)/4\pi, extending thus the result obtained by W. Mueller for the compact case. Concerning the latter, we prove that the sequence {\tau_n(M; \eta)} determines the complex-length spectrum of the manifold up to complex conjugation.