Totally non-congruence Veech groups

Abstract

Veech groups are discrete subgroups of $\mathrm{SL}(2,\mathbb{R})$ which play an important role in the theory of translation surfaces. For a special class of translation surfaces called origamis or square-tiled surfaces, their Veech groups are subgroups of finite index of $\mathrm{SL}(2,\mathbb{Z})$. We show that each stratum of the space of translation surfaces contains infinitely many origamis whose Veech group is a totally non-congruence group, i.e., it surjects to $\mathrm{SL}(2,\mathbb{Z}/n\mathbb{Z})$ for any $n$.