Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifolds

Motivated by Selberg’s 3/16 theorem for congruence covers of the modular surface, there have been many recent developments starting with Bourgain-Gamburd-Sarnak’s work to obtain analogous results for infinite volume hyperbolic manifolds. One such result by Oh-Winter is a generalization of the uniform exponential mixing formulation of Selberg’s theorem for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but also have a wide range of applications including 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.