Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces

Abstract

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma\backslash \mathbb{H}^2$ 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) = \Gamma_0(p)\backslash \mathbb{H}^2$ of $X$ for "almost" all primes $p$, provided the limit set of $\Gamma$ is thick enough.