  Let $\Gamma$ be a convex co-compact discrete group of isometries of the
hyperbolic plane $\mathbb{H}^2$, and $X=\Gamma\backslash \mathbb{H}^2$ 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 $\Gamma$. Using various techniques of thermodynamical formalism and
representation theory, we prove two new existence results of "sharp non-trivial
resonances" close to $\Re(s)=\delta_\Gamma$, both in the large degree limit,
for abelian covers and also infinite index congruence subgroups of
$SL2(\mathbb{Z})$.
