Congruence Veech groups (Q312326)

A translation surface is a surface \(X\) which one obtains from identifying the sides of a polygon in the Euclidean plane by translations. Equivalently it is a Riemann surface together with a holomorphic 1-form. If \(X\) is a translation surface then its Veech group \(\Gamma(X)\) is the Fuchsian group which is the image in \(\text{PSL}(2,\mathbb{R})\) of the subgroup of transformations \(g\in\text{SL}(2,\mathbb R)\) such that \(g\cdot X\) is biholomorphic as a translation surface to \(X\). A translation surface that does not admit a translation covering of a dgree \(d>1\) is called primitive. G. Weitze-Schmithüsen showed that many congruence subgroups of \(\text{PSL}(2,\mathbb{Z})\) can be realized as Veech groups of translation coverings of the once-punctured forms. Here, the author extends the definition of congruence groups to subgroups of the Veech groups \(\Gamma(X)\) for primitive translation surfaces \(X\) using their action on the homology with entries in \(\mathbb{Z}/a\mathbb{Z}\). Further she introduces a congruence level definition and a property that guarantees that partition stabilizing congruence subgroups occur as Veech groups of a translation covering. This property is satisfied for primitive translation surfaces with exactly one singular point. Finally, for the primitive translation surface glued together from two regular \(n\)-gons, \(n\) odd, she introduces a level, called generalized Wohlfart level, of subgroups of its Veech group and determines the connection to the Wohlfart level for congruence subgroups in \(\text{PSL}(2,\mathbb{Z})\).