Let (\Sigma,g) be a closed Riemannian surface of genus G with Anosov geodesic flow. Given an irreducible finite dimensional representation \rho of the fundamental group of its unit tangent bundle M, we can consider the twisted Ruelle zeta function \zeta_{\rho}(s). The zeta function \zeta_{\rho}(s) admits a meromorphic extension to the whole complex plane whose poles and zeros can be computed from the (twisted) Pollicott--Ruelle spectrum of (g,\rho).
We show that for a generic choice of \rho, \zeta_\rho vanishes at s=0 to order \dim(\rho)(2G-2) if \rho factors through \pi_1(\Sigma) and does to vanish otherwise. In the second case, we further show that \zeta_\rho(0) is given by the Reidemeister–Turaev torsion, thus extending Fried’s conjecture to a generic set of acyclic (but not necessarily unitary) representations. In higher dimensions, we compute the order of vanishing of the untwisted zeta function in an open and dense subset of Anosov metrics in the connected component of a hyperbolic 3-metric.
This is joint work with Zhongkai Tao.
