Large covers and sharp resonances of hyperbolic surfaces

Let $\mathrm{\Gamma }$ be a convex co-compact discrete group of isometries of the hyperbolic plane ${\mathbb{H}}^2$, and $X=\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^2$ the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian for large degree covers of $X$ given by a finite index normal subgroup of $\mathrm{\Gamma }$. Using various techniques of thermodynamical formalism and representation theory, we prove two new existence results of “sharp non-trivial resonances” close to $\mathrm{\Re }(s)={\delta }_{\mathrm{\Gamma }}$, both in the large degree limit, for abelian covers and also infinite index congruence subgroups of $SL2(\mathbb{Z})$.