Rationality of the SL(2,C)-Reidemeister torsion in dimension 3

TL;DR

This paper investigates the algebraic properties of the SL(2,C)-Reidemeister torsion in hyperbolic 3-manifolds, showing it is algebraic over certain fields and related to trace functions, with implications for the Volume Conjecture.

Contribution

It proves the torsion lies in at most quadratic extensions of the invariant trace field and establishes polynomial relations with trace functions, advancing understanding of torsion's algebraic nature.

Findings

Torsion is contained in at most quadratic extension of the invariant trace field.

Existence of polynomial relations between torsion and trace of meridian or longitude.

Coefficients of asymptotic expansions are elements of the field of rational functions on the character variety.

Abstract

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $\rho_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $\rho_0$ in $X_M$ and observed that it is an analytic map, which is the germ of a unique rational function on $X_M$. In the present paper we prove that (a) the torsion of a representation lies in at most quadratic extension of the invariant trace field of the representation, and (b) the existence of a polynomial relation of the torsion of a representation and the trace of the meridian or the longitude. We postulate that the coefficients of the $1/N^k$-asymptotics of the Parametrized Volume Conjecture for $M$ are elements of the field of rational functions on $X_M$.