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. 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 for convex cocompact hyperbolic surfaces. these are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the laplacian, affine sieve, and prime geodesic theorems. i will present a further generalization to higher dimensions and some of these immediate consequences.