Non-congruence of homology Veech groups in genus two

TL;DR

This paper investigates the homology Veech groups of genus two square-tiled surfaces, revealing they are mostly totally non-congruence subgroups, with symmetry properties contrasting previous results on Veech groups.

Contribution

It extends Weitze-Schmith"usen's results to homology Veech groups, showing they are predominantly totally non-congruence subgroups with symmetric properties.

Findings

Homology Veech groups are totally non-congruence subgroups.

Exceptions occur only at prime ideals above 2.

Results contrast with asymmetry in Veech groups regarding spin structure.

Abstract

We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of $\textrm{SL}_2(\mathcal{O}_D)$ where $\mathcal{O}_D$ is a quadratic order of square discriminant. Extending a result of Weitze-Schmith\"usen we show that also the homology Veech group is a totally non-congruence subgroup with exceptions stemming only from the prime ideals lying above 2. While Weitze-Schmith\"usen's result for Veech groups is asymmetric with respect to the spin structure our use of the homology Veech group yields a completely symmetric picture.