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An algorithm for finding the Veech group of an origami. (English) Zbl 1078.14036

Let us fix a once-punctured torus \(E^{*}\). An origami \({\mathcal O}\) is provided by a unramified covering \(p:X \to E^{*}\), where \(X\) is an analytically finite Riemann surface. Equivalently, if \(F_{2}\) is a given free group of rank \(2\) in the conformal automorphism group of the hyperbolic plane \({\mathbb H}^{2}\) so that \({\mathbb H}^{2}/F_{2}\) is a once-punctured torus, then an origami is given by the choice of a finite index subgroup \(H\) of \(F_{2}\). This produces a holomorphic embedding of \({\mathbb H}^{2}\) (the Teichmüller space of \(E^{*}\)) into the Teichmüller space of \(X\). The stabilizer of such an embedding in the modular group of \(X\) is the Veech group \(\Gamma({\mathcal O})\) associated to the origami \({\mathcal O}\). In this article, the author provides an algorithm which permits to compute \(\Gamma({\mathcal O})\).

MSC:

14H30 Coverings of curves, fundamental group
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
37F99 Dynamical systems over complex numbers
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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