Congruence Veech Groups

Abstract

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive translation surface which we call property $(\star)$. It guarantees that partition stabilising congruence subgroups of this level occur as Veech group of a translation covering. Each primitive surface with exactly one singular point has property $(\star)$ in every level. We additionally show that the surface glued from a regular $2n$-gon with odd $n$ has property $(\star)$ in level $a$ iff $a$ and $n$ are coprime. For the primitive translation surface glued from two regular $n$-gons, where $n$ is an odd number, we introduce a generalised Wohlfahrt level of subgroups in its Veech group. We determine the relationship between this Wohlfahrt level and the congruence level of a congruence group.