Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$

Abstract

Let $\Gamma<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $\Gamma(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(\Gamma (q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term for all square-free $q$ with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on $\Gamma (q) \backslash \mathbb{H}^2$ over $q$ as above. Our approach is to extend Dolgopyat's dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.