Twisted Pollicott--Ruelle resonances and zeta function at zero on surfaces

Abstract: For an orientable closed surface $(Σ,g)$ 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 $ρ\in U_g$, the twisted Ruelle zeta function $ζ{g,ρ}(s)$ vanishes at $s=0$ to order ${\rm dim}(ρ)(2G-2)$ if $ρ$ factors through $π_1(Σ)$, and does not vanish otherwise. In the second case, we show that $ζ{g,ρ}(0)$ is given by the Reidemeister--Turaev torsion, thus extending Fried's conjecture to a generic set of acyclic (but not necessarily unitary) representations. Our proof relies on computing the dimensions of the spaces of generalized twisted Pollicott--Ruelle resonant states at zero for any $ρ\in U_g$.