Spectral gaps and abelian covers of convex co-compact surfaces

Abstract

Given a convex co-compact hyperbolic surface $X=Γ\backslash \mathbb{H}^2$, we investigate the resonance spectrum $\mathcal{R}_j$ of the laplacian $Δ_j$ on large finite abelian covers $X=Γ_j\backslash \mathbb{H}^2$, where $Γ_j$ is a finite index normal subgroup of $Γ$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. We show that there exists an $\varepsilon>0$, such that for all $j$, resonances $\mathcal{R}_j$ in $\{ δ-\varepsilon< \mathrm{Re}(s) \leq δ\}$ are all real and satisfy a Weyl law given by the degree of the cover i.e. $\vert Γ/ Γ_j\vert$. 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 .