Higher‐dimensional Reidemeister torsion invariants for cusped hyperbolic 3‐manifolds

By P. Menal-Ferrer, J. Porti. 2011

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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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ca21a4395b3dcc0dc25d94ea70f7bc9dbaf47251