复双曲Klein群刚性问题的研究

The rigidity problem is an important research subject in the theory of Kleinian groups. Recently, there are a series of results on Mostow-Sullivan, lattice rigidities and the algebraic convergence of Kleinian groups. Based on our previous work, we will mainly discuss the rigidity of complex quasi-Fuchsian groups, and geometric and strong convergence of Kleinian groups and negatively curved groups in this project. The detailed research contents are as follows: (1) studying the Hausdorff measures of limit sets, critical exponents and the trace fields of complex quasi-Fuchsian groups, and giving some rigidity results for complex quasi-Fuchsian groups; (2) discussing the properties of limit groups of geometric convergence, strong convergence and polyhedron convergence in higher dimensional complex Kleinian groups and.negatively curved groups, and the relationship between various convergences with the help of relevant theories of Lie groups, Lie algebras and convergence groups; (3) exploring applications of rigidity theory of Kleinian groups to hyperbolic and negative curved manifolds. The research results of this project will reveal the behaviour of disctete groups acting on negatively curved spaces and provide some new ideas in further discussing the local structures and deformation theory of complex hyperbolic manifolds.

刚性问题是Klein理论研究中的重要课题。近年来，关于Mostow-Sullivan刚性、双曲格局部刚性以及Klein群的代数收敛性方面已有一系列的研究成果。 本项目旨在前期工作的基础上，围绕复拟Fuchs群的刚性，高维复双曲Klein群、负曲率群的几何收敛性和强收敛性方面展开研究。具体研究内容包括: (1) 研究复拟Fuchs群极限集的Hausdorff测度、临界指数以及群的迹生成的数域，给出复拟Fuchs群的一些刚性结果;（2）借助李群、李代数和收敛群的相关理论，讨论高维复双曲Klein群、负曲率群的几何收敛、强收敛和多面体收敛中极限群的性质以及各种收敛性之间的关系;（3）探讨Klein群刚性理论在双曲流形、负曲率流形上的应用。本项目的研究成果，有助于揭示不连续群在负曲率空间中作用的性态,为进一步讨论复双曲流形的局部结构和形变理论提供了新思路。

刚性问题是Klein理论研究中的重要课题。本项目在前期工作的基础上，围绕复拟Fuchs群的刚性，高维复双曲Klein群、负曲率群的离散性准则以及双曲型度量的应用方面展开研究。具体研究内容包括: (1) 研究复双Klein群，四元数Klein群以及无穷维双曲Klein群的Fuchs刚性结果;（2）借助李群、李代数和收敛群的相关理论，讨论负曲率群的离散性准则;（3）探讨双曲型度量在双曲流形、负曲率流形以及函数空间上的应用。本项目的研究成果，有助于揭示不连续群在负曲率空间中作用的性态,为进一步讨论复双曲流形的局部结构和形变理论提供了新思路。