Congruence subgroups and maximal Riemann surfaces
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- Schmutz, P. J Geom Anal (1994) 4: 207. doi:10.1007/BF02921547
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A global maximal Riemann surface is a surface of constant curvature −1 with the property that the length of its shortest simple closed geodesic is maximal with respect to all surfaces of the corresponding Teichmüller space. I show that the Riemann surfaces that correspond to the principal congruence subgroups of the modular group are global maximal surfaces. This result provides a strong geometrical reason that the Selberg conjecture, which says that these surfaces have no eigenvalues of the Laplacian in the open interval (0, 1/4), is true.