Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifolds

Abstract

Let $\Gamma$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $\Gamma \backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of Oh-Winter who proved the result for the $n = 2$ case. Following their approach, we use Dolgopyat's method for the proof of exponential mixing of the geodesic flow. We do this uniformly over congruence covers by establishing uniform spectral bounds for the congruence transfer operators associated to the geodesic flow. This requires another key ingredient which is the expander machinery due to Bourgain-Gamburd-Sarnak extended by Golsefidy-Varj\'{u}.