Totally non congruence Veech groups

TL;DR

This paper demonstrates that within every stratum of translation surfaces, there exist infinitely many origamis whose Veech groups are totally non-congruence, meaning they surject onto SL(2, Z/nZ) for all n, highlighting a rich diversity of such groups.

Contribution

It proves the existence of infinitely many origamis with totally non-congruence Veech groups in each stratum of translation surfaces.

Findings

Existence of infinitely many origamis with totally non-congruence Veech groups.

Veech groups surject onto SL(2, Z/nZ) for all n.

Presence of such groups in every stratum of translation surfaces.

Abstract

Veech groups are discrete subgroups of SL(2, 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 SL(2, 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 SL(2, Z/nZ) for any n.