Explicit spectral gap for Schottky subgroups of $\mathrm{SL}(2,\mathbb{Z})$

Abstract

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}(2,\mathbb{Z})$ and let $X$ be the corresponding hyperbolic surface. Provided that the limit set of $\Gamma$ is thick enough, we give an explicit interval $I_{\Gamma }=[0,a_{\Gamma }]$ such that for most congruence coverings $X_n$ of $X$, the eigenvalues in $I_{\Gamma }$ of the Laplace–Beltrami operators of $X_n$ and $X$, as well as their multiplicities, coincide.