The Galois action on M-Origamis and their Teichm\"uller curves

TL;DR

This paper studies a special class of translation surfaces called M-Origamis, exploring their Galois action, geometric structures, and symmetries, providing new proofs and explicit examples of Galois orbits and Teichmüller curves.

Contribution

It offers a new combinatorial proof of Galois faithfulness on M-Origamis' Teichmüller curves and extends understanding of their Galois action and symmetry groups.

Findings

Determined Strebel directions and cylinder decompositions of M-Origamis.

Identified the Veech group containing mma(2) and related symmetries.

Provided explicit examples of Galois orbits of M-Origamis.

Abstract

We consider a rather special class of translation surfaces (called M-Origamis in this work) that are obtained from dessins by a construction introduced by Martin M\"oller. We give a new proof with a more combinatorial flavour of M\"oller's theorem that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the Teichm\"uller curves of M-Origamis and extend his result by investigating the Galois action in greater detail. We determine the Strebel directions and corresponding cylinder decompositions of an M-Origami, as well as its Veech group, which contains the modular group $\Gamma(2)$ and is closely connected to a certain group of symmetries of the underlying dessin. Finally, our calculations allow us to give explicit examples of Galois orbits of M-Origamis and their Teichm\"uller curves.