  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.
