Gabriela Schmithüsen

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We study the Veech group of an origami, i.e. of a translation surface, tessellated by parallelograms. We show that it is isomorphic to the image of a certain subgroup of Aut(F2) in SL2(Z) = Out (F2). Based on this we present an algorithm that determines the Veech group. 1. Origamis as Teichmüller curves (Oriented) origamis (as defined in section 2.1) can be(More)
An origami is a combinatorial object (see section 2) that defines a translation surface (i.e. all transition maps are translations) of some genus g. Using this one can construct a certain affine complex curve in the moduli space Mg of regular projective complex curves of genus g. This curve is a Teichmüller curve, i.e. the image of a complex geodesic in the(More)
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(More)
In this chapter, we give an introduction to the theory of dessins d'enfants. They provide a charming concrete access to a special topic of arithmetic geometry: Curves defined over number fields can be described by such simple combina-torial objects as graphs embedded into topological surfaces. Dessins d'enfants are in some sense an answer of Grothendieck to(More)
We study an example of a Teichmüller curve CS in the moduli space M2 coming from an origami S. It is particular in that its points admit V4 as a subgroup of the automorphism group. We give an explicit description of its points in terms of a ne plane curves, we show that CS is a nonsingular, a ne curve of genus 0 and we determine the number of cusps in the(More)
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