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

2009/08/12 by Jérôme Dubois, Dubois, Jerome, Stavros Garoufalidis +1
Mathematics · #57N10 (Primary) 57M25 (Secondary) #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.0908.1690

openalex publication_date 2009/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ρ₀. Porti defined a non-abelian Reidemeister torsion in a neighborhood of ρ₀ 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.

Citations

• Chern-Simons theory, analytic continuation and arithmetic

Related

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