Spectral gaps and abelian covers of convex co-compact surfaces

Given a convex co-compact hyperbolic surface $X=\Gamma\backslash \mathbb{H}^2$, we investigate the resonance spectrum $\mathcal{R}_j$ of the laplacian $\Delta_j$ 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 show that there exists an $\varepsilon>0$, such that for all $j$, resonances $\mathcal{R}_j$ in $\{ \delta-\varepsilon< \mathrm{Re}(s) \leq \delta \}$ are all 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 estimates for transfer operators. One of the new ingredients of the proof is the decay of oscillatory integrals with respect to Patterson-Sulivan measures, obtained recently by Bourgain-Dyatlov arXiv:1704.02909 .