Generic twisted Pollicott--Ruelle resonances and zeta function at zero

Abstract

For a connected orientable closed surface $(\Sigma,g)$ of genus $G$ with Anosov geodesic flow, we show the existence of an open subset $U_g$ of finite-dimensional irreducible representations of the fundamental group of its unit tangent bundle, whose complement has complex codimension at least one and such that for any $\rho \in U_g$, the twisted Ruelle zeta function $\zeta_{g,\rho}(s)$ vanishes at $s=0$ to order ${\rm dim}(\rho)(2G-2)$ if $\rho$ factors through $\pi_1(\Sigma)$, and does not vanish otherwise. In the second case, we show that $\zeta_{g,\rho}(0)$ is given by the Reidemeister--Turaev torsion, thus extending Fried's conjecture to a generic set of acyclic (but not necessarily unitary) representations. We also show that the order of vanishing of the untwisted zeta function is constant for an open and dense subset of Anosov metrics in the connected component of a hyperbolic $3$-metric. Our proofs rely on computing the dimensions of the spaces of generalized twisted Pollicott--Ruelle resonant states at zero.