Calculation of Veech groups and Galois invariants of general origamis

TL;DR

This paper develops an algorithm to compute Veech groups and Galois invariants of general origamis, extending analysis beyond pure half-translation structures, and provides comprehensive calculations for origamis up to degree 7.

Contribution

It introduces a novel algorithm for calculating Veech groups of general origamis and computes their Galois invariants for degrees up to 7.

Findings

Calculated equivalence classes and orbits for origamis of degree ≤ 7.

Developed an algorithm for Veech group computation of general origamis.

Analyzed Galois invariants for a broad class of origamis.

Abstract

Nontrivial examples of Teichm\"uller curves have been studied systematically with notions of combinatorics invariant under affine homeomorphisms. An origami (square-tiled surface) induces a Teichm\"uller 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 $d\leq 7$.