Explicit spectral gaps for random covers of Riemann surfaces

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=Γ\backslash\mathbb{H}$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. 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 $ε>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=σ+it$ of $X_{n}$ with $σ\in[\frac{3}{4}δ+ε,δ]$ and $t\in[-H,H]$. This implies in the case of $δ>\frac{1}{2}$ 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 $η=η(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>δ-η\,\}$.

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