复双曲格相关问题的研究

The study on complex hyperbolic lattices is one of the central topics in complex hyperbolic geometry. It is closely related to many areas, e.g. Lie group,complex geometry,dynamic systerm. The purpose of this plan is twofold. First, we consider two Fuchsian triangle groups or two lattices in PSL(2,R), embed them in the isometric group of complex hyperbolic space, PU(2,1) and study that in which cases the resulting hybrid is discrete. Secondly, we will discuss the simple algorithm for obtaining the generators of Picard modular groups in complex two dimensions or complex three dimensions. We also study the fundamental domain, volume of the Eisenstein-Picard modular group in complex three dimensions.

复双曲格是复双曲几何中的一个主要研究内容，它与李群，复几何，动力系统等数学分支的研究有着紧密的联系.本项目将主要研究复双曲格的几何性质. 首先，我们将研究两个Fuchsian三角群或两个PSL(2,R)中的格到复双曲等距群PU(2,1)中的嵌入，得到Hybrid的离散条件.然后，我们将讨论复二维和复三维Picard模群的生成子，构造复三维Eisenstein-Picard模群的基本域，并估计复双曲格的体积.

本项目主要研究了一些复双曲等距子群相关的几何，代数方面的性质。在以下几个方面取得了一些成果：高维Picard 模群的生成子问题；PU(2,1)的包含螺旋抛物元素子群的离散准则； 给出了高维复双曲等距群SU(n,1)的Fuchsian子群的特征；给出了PU(2,1)中两个Heisenberg传递生成的离散自由子群的条件；得到了任意维复双曲流形中嵌入球的半径的大小的估计。