Study on Fundamental Regions for 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。在Mobius变换的情况下，Jorgensen的不等式是非常著名的。Kamiya，BasMajian-Miner和Parker讨论了PU(1，2；C)的子群具有抛物线元素的离散条件。我们给出了它们之间的关系，并将BasMajian-Miner稳定盆定理推广到PU(1，2；C)的更广泛的子群上。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不等式的类比，其中一个元素是斜行的或边界椭圆的。2)定义了PU(1，n；C)的一个元素的广义等距。利用PU(1，n；C)的离散子群的元素的广义等距球面，我们构造了一个基本域，它是Ford域的推广。并证明了这种广义Ford域与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变形空间。

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.