Finitely Maximal Fuchsian Groups
Finitely Maximal Fuchsian Groups
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有限极大 Fuchsian 群
DOI：
10.1112/jlms/s2-6.1.29
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发表时间：
1972
影响因子：
1.2
通讯作者：
D. Singerman
中科院分区：
文献类型：
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作者：
D. Singerman
对于 Fuchsian 群 F，我们指的是 PSL (2, U) 的有限生成离散子群，即上半平面 U 的所有共形同胚的群。Greenberg [3] 定义 Fuchsian 群为有限极大，如果不存在另一个包含有限索引的 Fuchsian 群。设R(T)表示所有同构r的集合：F-+PSL(2,R)，其性质是r(F)是离散的并且r将抛物线(双曲边界)元素映射到抛物线(双曲边界)元素。格林伯格证明，对于某些 Fuchsian 群 F，存在一个包含它且具有有限索引的 Fuchsian 群 Fo，使得每个 r e R (T) 只是一个 r0 eR (T0) 对 F 的限制。在这种情况下，r (F) 永远不是有限最大值。（他还表明，对于其他群 F，r (F) “通常”是有限最大值）。格林伯格确定了其中一些对 F、Fo。在本文中，我们确定所有此类对。
By a Fuchsian group F we shall mean a finitely generated discrete subgroup of PSL (2, U), the group of all conformal homeomorphisms of the upper half-plane U. Greenberg [3] defines a Fuchsian group to be finitely maximal if there does not exist another Fuchsian group containing it with finite index. Let R (T) denote the set of all isomorphisms r: F-+ PSL (2, R) with the property that r (F) is discrete and r maps parabolic (hyperbolic boundary) elements to parabolic (hyperbolic boundary) elements. Greenberg showed that for some Fuchsian groups F there exists a Fuchsian group Fo containing it with finite index, such that every r e R (T) is just the restriction to F of an r0 eR (T0). In this case r (F) is never finitely maximal.(He also showed that for other groups F, r (F) was" usually" finitely maximal). Greenberg determined some of these pairs F, Fo. In this paper we determine all such pairs.