Explicit spectral gaps for random covers of Riemann surfaces

Abstract

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $\epsilon>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=\sigma+it$ of $X_{n}$ with $\sigma\in[\frac{3}{4}\delta+\epsilon,\delta]$ and $t\in[-H,H]$. This implies in the case of $\delta>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $\eta=\eta(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>\delta-\eta\,\}$.