The Fried conjecture in small dimensions

We study the twisted Ruelle zeta function ζ X ( s ) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove the Fried conjecture, relating Reidemeister torsion and ζ X ( 0 ) . In higher dimensions, we show more generally that ζ X ( 0 ) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show the Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where the Fried conjecture holds true.