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Origamis with non congruence Veech groups

By Gabriela Schmithüsen


In this article we give an introduction to origamis (often also called square-tiled surfaces) and their Veech groups. As main theorem we prove that in each genus there exist origamis, whose Veech groups are non congruence subgroups of SL2(Z). The basic idea of an origami is to obtain a topological surface from a few combinatorial data by gluing finitely many Euclidean unit squares according to specified rules. These surfaces come with a natural translation structure. One assigns in general to a translation surface a subgroup of GL2(R) called the Veech group. In the case of surfaces defined by origamis, the Veech groups are finite index subgroups of SL2(Z). These groups are the objects we study in this article. One motivation to be interested in Veech groups is their relation to Teichmüller disks and Teichmüller curves, see e.g. the article [H 06] of F. Herrlich in the same volume: A translation surface of genus g defines in a geometric way an embedding of the upper half plane into the Teichmüller space Tg of closed Riemann surfaces of genus g. The image is called Teichmüller disk. Its projection to the moduli space Mg is sometimes a complex algebraic curve, called Teichmülle

Year: 2007
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