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 $X=\Gamma \backslash \mathbf {H}$ . 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 $\to 1$ as $n\to \infty $ , there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>\delta -\eta \,\}$ .