i will report on work in progress showing that the geodesic flow on any geometrically finite, rank one, locally symmetric space is exponentially mixing with respect to the bowen-margulis-sullivan measure of maximal entropy. the method is coding-free and is instead based on a spectral study of transfer operators on suitably constructed anisotropic banach spaces, ala gouezel-liverani, to take advantage of the smoothness of the flow. as a consequence, we obtain more precise information on the size of the essential spectral gap as well as the meromorphic continuation properties of laplace transforms of correlation functions.