Totally non congruence Veech groups

Non-congruence of homology Veech groups in genus two

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-Schmithüsen 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-Schmithüsen's result for Veech groups is asymmetric with respect to the spin structure our use of the homology Veech group yields a completely …

Congruence Veech Groups

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 additi…

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

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.

具有非同余Veech群的Origamis

Origamis with non congruence Veech groups

作为主要结果，我们证明了对任意g > 1，存在亏格g的平移曲面，其Veech群是SL（2，Z）的非同余子群。我们使用origamis/正方形平铺曲面来生成示例。本文分为两部分：第一部分介绍了平移曲面、origamis、Veech群和Teichmueller曲线，并证明了亏格2中的两个origamis的Veech群是非同余群;第二部分给出了一种生成Veech群递减的origamis序列的方法。这是用来证明主要结果的。

Regular origamis with totally non-congruence groups as Veech groups

Veech groups are an important tool to examine translation surfaces and related mathematical objects. Origamis, also known as square-tiled surfaces, form an interesting class of translation surfaces with finite index subgroups of SL(2,Z) as Veech groups. We study when Veech groups of origamis with maximal symmetry group are totally non-congruence groups, i.e., when they surject onto SL(2, Z/nZ) for each natural number n. For this, we use a result of Schlage-Puchta and Weitze-Schmithüsen to deduce sufficient conditions on the deck transformation group of the origami. More precisely, we show that…

不含抛物元的Veech群

Veech groups without parabolic elements

本文证明了一个"bouillabaisse"曲面（具有两个横向抛物元素的平移曲面）具有全实迹场。作为推论，不存在没有抛物元的非平凡Veech群。证明遵循Veech的观点瑟斯顿的建设的伪Anosov同构。