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arXiv · 2006.00905

Calculation of Veech groups and Galois invariants of general origamis

Shun Kumagai
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Abstract

Nontrivial examples of Teichmüller curves have been studied systematically with notions of combinatorics invariant under affine homeomorphisms. An origami (square-tiled surface) induces a Teichmüller curve for which the absolute Galois group acts on the embedded curve in the moduli space. In this paper, we study general origamis not admitting pure half-translation structure. Such a flat surface is given by a cut-and-paste construction from origami that is a translation surface. We present an algorithm for the simultaneous calculation of the Veech groups of origamis of given degree. We have calculated the equivalence classes, the $PSL(2,\mathbb{Z})$-orbits, and some Galois invariants for all the patterns of origamis of degree $d\leq 7$.

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Shun Kumagai. 2023-06-07. Calculation of Veech groups and Galois invariants of general origamis. https://arxiv.org/abs/2006.00905

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