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Infinitely generated Veech groups. (English) Zbl 1056.30044

Veech groups are subgroups of finite index in a Teichmüller modular group. They are subgroups of \(SL(2,\mathbb R)\). The construction is due to Thurston who investigated surface diffeomorphisms by way of locally flat structures. A {translation surface} is a Riemann surface, where, apart from finite set of points ({singularities}), the transition functions are translations. The first main theorem of this paper is that there are translation surfaces whose Veech group is not finitely generated but is Fuchsian of the first kind. This answers a question posed by W. Veech. The paper is devoted more generally to a detailled analysis of the geometry and group theory of Veech groups.

MSC:

30F30 Differentials on Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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