The Journal of Geometric Analysis

, Volume 4, Issue 2, pp 207–218

Congruence subgroups and maximal Riemann surfaces


DOI: 10.1007/BF02921547

Cite this article as:
Schmutz, P. J Geom Anal (1994) 4: 207. doi:10.1007/BF02921547


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.

Math Subject Classification

30F 20H05 53C22 

Key Words and Phrases

Riemann surfaces congruence subgroups geodesics systoles Selberg’s 1/4 conjecture 

Copyright information

© Mathematica Josephina, Inc. 1994

Authors and Affiliations

  1. 1.Mathematisches InstitutETH-ZentrumZürichSwitzerland

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