Calculation of Veech groups and Galois invariants of general origamis

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,)-orbits, and some Galois invariants for all the patterns of origamis of degree d ≤ 7.