Abstract:
Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing of the geodesic flow. Bourgain-Gamburd-Sarnak’s breakthrough works initiated many recent developments to generalize Selberg’s theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing of the geodesic flow for congruence covers of convex cocompact hyperbolic surfaces. We present a further generalization to higher dimensions for the frame flow which is a new result even for a single manifold. Immediate applications include an asymptotic formula for matrix coefficients with an exponential error term, exponential equidistribution of holonomy of closed geodesics, affine sieve, and a uniform resonance-free half plane for the resolvents of the Laplacians.