Congruence subgroups and maximal Riemann surfaces (Q1329634)

A global maximal Riemann surface is a surface of constant curvature \(-1\) with the property that length of its shortest closed geodesic is maximal with respect to all surfaces of the corresponding Teichmüller space. The author shows that the Riemann surfaces corresponding to the principal congruence subgroups of the modular group are global maximal surfaces. Some questions related to the Selberg conjecture, which says that the above mentioned Riemann surfaces have no eigenvalues of the Laplacian in the open interval \((0,1/4)\) are discussed.