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Given a convex co-compact hyperbolic surface $X=\Gamma\backslash\mathbb{H}^2$, we investigate the resonance spectrum $\mathcal{R}_j$ of thelaplacian $\Delta_j$ on large finite abelian covers $X=\Gamma_j\backslash\mathbb{H}^2$, where $\Gamma_j$ is a finite index normal subgroup of $\Gamma$.Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We showthat there exists an $\varepsilon>0$, such that for all $j$, resonances$\mathcal{R}_j$ in $\{ \delta-\varepsilon< \mathrm{Re}(s) \leq \delta \}$ areall real and satisfy a Weyl law given by the degree of the cover i.e. $\vert\Gamma/ \Gamma_j\vert$. In particular, we prove that for large imaginary parts,there is a uniform resonance gap, obtained through uniform Dolgopyat estimatesfor transfer operators. One of the new ingredients of the proof is the decay ofoscillatory integrals with respect to Patterson-Sulivan measures, obtainedrecently by Bourgain-Dyatlov arXiv:1704.02909 .

Spectral gaps and abelian covers of convex co-compact surfaces