Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Let $\mathrm{\Gamma }$ be a Schottky subgroup of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ and let $X=\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a uniform and explicit spectral gap for the Laplacian on the Hecke congruence covers $X_0(p)={\mathrm{\Gamma }}_0(p)\mathrm{\backslash }{\mathbb{H}}^2$ of $X$ for “almost” all primes $p$, provided the limit set of $\mathrm{\Gamma }$ is thick enough.