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

Abstract

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL} (2,\mathbb{Z})$. We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface $\Gamma \backslash \mathbb{H}^2$ provided the limit set of $\Gamma$ is thick enough.