Regular origamis with totally non-congruence groups as Veech groups

TL;DR

This paper investigates when the Veech groups of certain origamis are totally non-congruence groups, providing conditions based on their deck transformation groups, especially focusing on quotients of triangle groups and Hurwitz groups.

Contribution

It establishes sufficient conditions for origamis with specific deck transformation groups to have Veech groups that are totally non-congruence groups.

Findings

Origamis with quotients of triangle groups as deck groups satisfy the conditions.

All Hurwitz groups are examples of such deck transformation groups.

Identifies when Veech groups are surjective onto SL(2, Z/nZ) for all n.

Abstract

Veech groups are an important tool to examine translation surfaces and related mathematical objects. Origamis, also known as square-tiled surfaces, form an interesting class of translation surfaces with finite index subgroups of SL(2,Z) as Veech groups. We study when Veech groups of origamis with maximal symmetry group are totally non-congruence groups, i.e., when they surject onto SL(2, Z/nZ) for each natural number n. For this, we use a result of Schlage-Puchta and Weitze-Schmith\"usen to deduce sufficient conditions on the deck transformation group of the origami. More precisely, we show that origamis with certain quotients of triangle groups as deck transformation groups satisfy this condition. All Hurwitz groups are such quotients.