On the monodromy and spin parity of single-cylinder origamis in the minimal stratum
In a paper with Menasco-Nieland, the first author constructed factorially many origamis in the minimal stratum of the moduli space of translation surfaces having simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these origamis were constructed using the minimal number of squares required for origamis in the minimal stratum. We shall call such origamis minimal $[1,1]$-origamis. In this work, we calculate all of the spin parities of the Aougab-Menasco-Nieland origamis, and we therefore determine the connected component of the minimal stratum within which each is contained. Motivated by understanding the $\SL(2,\Z)$-orbits of these origamis, we investigate their monodromy groups, in particular proving that all of them are alternating or projective special linear groups. In fact, we prove more generally that the monodromy group of a minimal $[1,1]$-origami must almost always be a finite simple group. Finally, we determine the Kontsevich-Zorich monodromies of these origamis in low genus and give a conjecture in general. Note that previous works in the literature (e.g., that of Eskin-Kontsevich-Zorich, Filip-Forni-Matheus, Gutiérrez-Romo, Kany-Matheus, Matheus-Yoccoz-Zmiaikou, and Zorich) often chose to discuss just one of these $\SL(2,\Z)$-invariants at a time: in particular, to the best our knowledge, this is one of the first places where all of these $\SL(2,\Z)$-invariants are computed explicitly in a single paper for such a large family of origamis.