On Discrete Subgroups of PU(1,2;C) with Heisenberg Translations
On Discrete Subgroups of PU(1,2;C) with Heisenberg Translations
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DOI：
10.1112/s0024610700001435
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发表时间：
2000-12
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通讯作者：
S. Kamiya
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作者：
S. Kamiya
在离散群的研究中，找到一个群是离散的条件是很重要的。给定一个包含一个不动点为∞的抛物元的Möbius变换的离散子群，一个经典的结果，称为清水引理，给出了群中不动点为∞的那些元素的等距圆半径的一致界。最近帕克[8]证明了如果PU（1，n; C）的离散子群G包含一个海森堡平移g，则G中任何不与g共享不动点的元素都有一个等距球面，其半径由g在其中心的平移长度的函数上界。帕克定理被认为是Shimizu引理的推广。在[1]中，Basmajian和Miner利用稳定盆定理分别得到了PU（1，2; C）的离散子群的定性相似结果。
In the study of discrete groups it is important to find conditions for a group to be discrete. Given a discrete subgroup of Möbius transformations containing a parabolic element with fixed point ∞, a classical result, called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker [8] has shown that if a discrete subgroup G of PU(1, n; C) contains a Heisenberg translation g, then any element of G not sharing a fixed point with g has an isometric sphere whose radius is bounded above by a function of the translation length of g at its centres. Parker's theorem is considered as a generalization of Shimizu's lemma. In [1] Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1, 2; C) by using their stable basin theorem.