Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

TL;DR

This paper establishes an explicit relationship between analytic and Reidemeister torsion for non-compact hyperbolic 3-manifolds with cusps, using special values of Ruelle zeta functions and recent mathematical advances.

Contribution

It provides the first explicit formula connecting analytic and Reidemeister torsion in the context of hyperbolic 3-manifolds with cusps, expanding understanding of their geometric invariants.

Findings

Derived an explicit formula relating the two torsions.

Expressed analytic torsion in terms of Ruelle zeta function values.

Utilized recent work of Pere Menal-Ferrer and Joan Porti.

Abstract

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.