Spectral gaps and abelian covers of convex co-compact surfaces

Abstract

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 $\epsilon >0$, such that for all $j$, resonances ${\mathcal{R}}_j$ in $\{\delta -\epsilon <\mathrm{Re}(s)\le \delta \}$ are all real and satisfy a Weyl law given by the degree of the cover i.e. $|\Gamma /{\Gamma }_j|$. 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 .

This is a follow up to arXiv:1710.05666