Symmetric Fuchsian quadrilateral groups and modular embeddings
Symmetric Fuchsian quadrilateral groups and modular embeddings
复制标题
对称紫红色四边形群和模块化嵌入
DOI：
10.1093/qjmath/53.1.75
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发表时间：
2002
影响因子：
0.7
通讯作者：
S. Ricker
中科院分区：
文献类型：
--
作者：
S. Ricker
Paula Cohen和Jürgen Wolfart的一个结果指出，所有Fuchsian三角形群和它们的有限指数子群都允许所谓的模嵌入，也就是说，它们可以被看作是算术群的一个子群，并且可以这样做，使得三角形群在上半平面上的作用与算术群在上半平面的乘积上的作用是全纯相容的。在本文中，我们关心的问题是否可以模仿对称的Fuchsian四边形群，即Fuchsian四边形群拥有一个基本的区域与一个额外的对称性。我们的主要结果是：这些群的模嵌入的存在性等价于某些扩展的Riemann映射的存在性，并且签名为[2，2，2，t]和[2，2，t，t]，t ∈ {3，4，6}的对称四边形群永远不允许模嵌入，除非它们本身已经是算术的。
A result of Paula Cohen and Jürgen Wolfart states that all Fuchsian triangle groups and their finite index subgroups admit a so‐called modular embedding, that is, they may be seen as a subgroup of an arithmetic group and this may be done in such a way that the action of the triangle group on the upper half‐plane is holomorphically compatible with the action of the arithmetic group on a product of upper half‐planes. In this paper we are concerned with the question whether this construction could be mimicked for symmetric Fuchsian quadrilateral groups, that is, Fuchsian quadrilateral groups which possess a fundamental region with an additional symmetric property. Our main result is that the existence of a modular embedding for those groups is equivalent to the existence of certain extended Riemann mappings and that symmetric quadrilateral groups of signature [2, 2, 2, t] and [2, 2, t, t], t ∈ {3, 4, 6} will never admit a modular embedding unless they are already arithmetic themselves.