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.

arXiv comment31 pages In this version we do minor changes to the introduction and correct several typos