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

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

2009-08-12

arxiv.org/abs/0908.1690v3

Mathematics

Geometric Topology

13 pages

Scientific paper

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 $X_M$.

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