Congruence subgroups and maximal Riemann surfaces |
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Authors: | Paul Schmutz |
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Affiliation: | 1. Mathematisches Institut, ETH-Zentrum, CH-8092, Zürich, Switzerland
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Abstract: | 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. |
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