Study on Fundamental Regions for Discrete Subgroups of PU(1,n;C)

1)在离散群的研究中，找到一个群是离散的条件是很重要的。在Mobius变换的情况下，Jorgensen不等式非常著名。Kamiya，Basmajian-Miner和帕克讨论了PU（1，2;C）中具有抛物元的子群的离散性条件。给出了它们之间的关系，并将Basmajian-Miner的稳定盆定理推广到PU（1，2;C）的更广的子群类.利用复双曲空间边界上的交比，我们得到了由两个元素生成的非初等群的Jorgensen不等式的类似结果，其中一个元素是斜驶的或边界椭圆的。2）我们定义了PU（1，n;C）中元素的广义等距。利用PU（1，n;C）的离散子群的元素的广义等距球面，构造了一个基本整环，它是福特整环的推广.并证明了这种广义福特整环与Dirichlet多面体之间的联系。现在我们正试图利用这些基本区域来研究形变空间。

1) In the study of discrete groups, it is important to find conditions for a group to be discrete. In the case of Mobius transformations Jorgensen's inequality is very famous. Kamiya, Basmajian-Miner and Parker have discussed the discreteness conditions for subgroups of PU(1,2;C) with parabolic elements. We give the relationship between them and also generalize Basmajian-Miner's stable basin theorem to wider class of subgroups of PU(1,2;C). By using the cross-ratio in the boundary of complex hyperbolic space, we obtain analogues of Jorgensen's inequality for non-elementary groups generated by two elements, one of which is either loxodromic or boundary elliptic.2) We define the generalized isometric of an element of PU(1,n;C). By using the generalized isometric spheres of elements of a discrete subgroup of PU(1,n;C), we construct a fundamental domain, which is regarded as a generalization of the Ford domain. And we show the connection between this generalized Ford domain and Dirichlet polyhedron. Now we are trying to use these fundamental regions for studying the deformation space.