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

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.