On the Critical Line of Convex Co-Compact Hyperbolic Surfaces

Abstract

Let Γ be a convex co-compact Fuchsian group. We formulate a conjecture on the critical line, i.e. what is the largest half-plane with finitely many resonances for the Laplace operator on the infinite-area hyperbolic surface \({X = \Gamma \backslash \mathbb{H}^2}\). An upper bound depending on the dimension δ of the limit set is proved which is in favor of the conjecture for small values of δ and in the case when δ > 1/2 and Γ is a subgroup of an arithmetic group. New omega lower bounds for the error term in the hyperbolic lattice point counting problem are derived.