Torsion function on character varieties

In this paper we define the Reidemeister torsion as a rational function on a geometric component of the character variety of a one-cusped hyperbolic manifold $M$. We study its poles and zeros and deduce that under some mild hypothesis on the manifold $M$ this function is non-constant, answering partially to a question addressed in the article "Twisted Alexander Polynomials of Hyperbolic Knots" of N. Dunfield, S. Friedl and N. Jackson.