THE TWISTED RUELLE ZETA FUNCTION ON COMPACT HYPERBOLIC ORBISURFACES AND REIDEMEISTER-TURAEV TORSION

- Let X be a compact hyperbolic surface with finite order singularities, X1 its unit tangent bundle. We consider the Ruelle zeta function R(s; p) associated to a representation p:pi 1(X1)-+ GL(V rho). If p does not factor through pi 1(X), we show that the value at 0 of the Ruelle zeta function equals the sign-refined Reidemeister-Turaev torsion of (X1, p) with respect to the Euler structure induced by the geodesic flow and to the natural homology orientation of X1. It generalizes Fried's conjecture to non-unitary representations, and solves the phase and sign ambiguity in the unitary case. We also compute the vanishing order and the leading coefficient of the Ruelle zeta function at s = 0 when p factors through x1(X).