Symmetries of geodesic flows on covers and rigidity

We define and study the foliated centralizer: the group of C^ centralizer elements of the lift of an Anosov system on a non-compact manifold which additionally preserve the stable and unstable foliations. When the Anosov system is the geodesic flow of a closed Riemannian manifold with pinched negative sectional curvatures, we prove some rigidity properties for the foliated centralizer of the lift of the flow to the universal cover: it is a finite-dimensional Lie group, which is moreover discrete (modulo the action of the flow itself) unless the metric is homothetic to some real hyperbolic metric. This result is inspired by the study of isometries of universal covers that appeared originally in the work of Eberlein, and later in Farb and Weinberger, as well as centralizer rigidity results in dynamics.