STUDY ON DISCRETE SUBGROUPS OF PU(1, n ; C)

1)在离散群的研究中，找到群是离散的条件是很重要的。给出了一个包含不动点为∞的抛物型元素的莫比乌斯变换的离散子群，一个经典的结果，称为清水引理，给出了该群中不固定∞的元素的等距圆半径的一致界.最近，Parker证明了：如果PU(1，n；C)的一个离散子群G包含一个Heisenberg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，它的半径是由g在其中心的平移长度的函数来定界的。该Parker定理被认为是清水引理的推广。BasMajian和Miner利用稳定盆定理，分别对PU(1，2；C)的离散子群得到了定性相似的结果。首先改进了稳定盆定理，然后证明了在一定条件下，Parker定理给出了具有Heisenberg平移的群的BasMajian和Miner的离散性条件。后者回答了Parker论文中提出的一个问题。2)设G是PU(1，n；C)的一个离散子群。对于Siegel域的边界点y，我们定义了PU(1，n；C)的元素f的广义等距球面I^y(F)。利用G的元素的广义等距球面，我们构造了G的一个基本域P^y(G)，它是Ford域的推广。证明了中心为w的Dirichlet多面体D(W)转换为P^y(G)为w→y。

1) In the study of discrete groups, it is important to find conditions for a group to be discrete. Given a discrete subgroup of Mobius transformations containing a parabolic element with fixed point ∞, a classical result, which is called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker 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 centers. This Parker's theorem is considered as a generalization of Shimizu's lemma. Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1,2 ; C) by using their stable basin theorem. First we improve the stable basin theorem, and second we show that under some conditions Parker's theorem yields the discreteness condition of Basmajian and Miner for groups with a Heisenberg translation. The latter answers a question posed in Parker's paper.2) Let G be a discrete subgroup of PU(1, n ; C). For a boundary point y of the Siegel domain, we define the generalized isometric sphere I^y (f) of an element f of PU(1, n ; C). By using the generalized isometric spheres of elements of G, we construct a fundamental domain P^y (G) for G, which is regarded as a generalization of the Ford domain. And we show that the Dirichlet polyhedron D(w) for G with center w convereges to P^y(G) as w → y.