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

Let $Λ$ 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 $X=Λ\backslash\H$ has infinitely many resonances in an effective strip depending on the dimension of the limit set $δ$. Applications to lower bounds for the hyperbolic lattice point counting problem are derived.

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