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 $PSL(2,\mathbf{Z})$-orbits, and some Galois invariants for all the patterns of origamis of degree $d \leq 7$.
Funding Statement
This work was supported by JSPS KAKENHI Grant Number JP21J12260.
Acknowledgment
The author thanks to Prof. Toshiyuki Sugawa and Prof. Hiroshige Shiga for their helpful advices and comments. The author thanks the anonymous referee for his careful reading of the manuscript and his suggestions for improvement. The computation was carried out using the computer resource offered under the category of General Projects by Cyber Science Center, Tohoku University.
Citation
Shun Kumagai. "Calculation of Veech groups and Galois invariants of general origamis." Kodai Math. J. 47 (2) 137 - 161, June 2024. https://doi.org/10.2996/kmj47202
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