Noncongruence subgroups in H(2)

TL;DR

This paper investigates the congruence properties of certain subgroups of the modular group, specifically those arising as Veech groups of square-tiled surfaces in genus two, to understand their structure and classification.

Contribution

It provides new insights into the congruence problem for Veech groups associated with genus two square-tiled surfaces, a previously less explored area.

Findings

Identifies which Veech groups are congruence subgroups.

Classifies noncongruence Veech groups in genus two.

Advances understanding of the structure of subgroups in the modular group.

Abstract

We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.