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 ∖ 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 → ∞ , there are no new resonances s = σ + i t of X n with σ ∈ [ 3 4 δ + ϵ , δ ] and t ∈ [ − H , H ] . This implies in the case of δ > 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 → 1 as n → ∞ , there are no new resonances of X n in the region { s : Re ( s ) > δ − η } .