On the resonances of convex co-compact subgroups of arithmetic groups

Let $\mathrm{\Lambda }$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface X=\Lambda \backslash\H has infinitely many resonances in an effective strip depending on the dimension of the limit set $\delta$. Applications to lower bounds for the hyperbolic lattice point counting problem are derived.