The deficiency of being a congruence group for Veech groups of origamis

TL;DR

This paper investigates how close Veech groups of origamis are to being congruence groups, introducing a measure of deficiency and demonstrating that certain origamis are maximally distant from being congruence groups.

Contribution

It defines a new measure for the deficiency of a subgroup from being a congruence group and applies it to Veech groups of origamis, showing they are often far from being congruence groups.

Findings

Veech groups of origamis in H(2) are far from congruence groups

The index of the image in SL(2,Z/nZ) is maximized at the Wohlfahrt level

Existence of infinite families of origamis maximally distant from congruence groups

Abstract

We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or square-tiled surfaces) in the stratum H(2) are far away from being congruence groups and that in each genus one finds an infinite family of origamis such that they are "as far as possible" from being a congruence group.