An algorithm for classifying origamis into components of Teichmüller curves

Shun Kumagai

Published 2020-05-29Version 1

Non trivial examples of Veech groups have been studied systematically with the notion of combinatorics coming from coverings. For abelian origamis, coverings of once punctured torus, their Veech groups are described by Schmith\"usen in terms of monodromy. Shinomiya applied her method for translation coverings of the surface obtained from regular $2n$-gon. Their results enable us to specify the Veech group in a concrete example by using the Reidemeister-Schreier method. In this paper, we deal with origamis including non abelian origamis using a method inspired to `comparisons of parallelogram decompositions'. Our algorithms classify all origamis of given degree into natural isomorphism classes and specify their Veech groups in parallel.