Geometric structures and surface group representations into Lie groups

拟议的项目旨在发展几何表示的基本群的表面S（表面组）到李群。PSL（2，R）和PSL（2，C）的连续生成离散群称为Kleinian群，特别是Kleinian群的研究已有几十年的历史。它们分别是二维和三维双曲空间的保向等距群，代表负弯曲几何。它们在泰希米勒理论、黎曼曲面的研究和三维双曲流形，特别是三维拓扑流形的几何化中扮演着重要角色。我提出的项目的主题是推广有趣的性质Kleinian群和Teichmüller理论，更一般的表示和相关的几何，拓扑，代数，分析和动力学。该建议的一个研究方向是理解曲面群的几何表示为PSL（2，R）或PSL（2，C），其图像是非离散子群（非离散表示）。这些项目寻求非离散表示的几何意义。特别地，它们与映射类群对特征变量的作用的动力学、表面群表示的空间密切相关。另一个研究方向是研究曲面群到高阶李群的表示，特别是PSL（3，R）和PSL（3，C）。当一阶李群对应于“负弯曲几何”时，高阶李群对应于“非正弯曲几何”并给出丰富的几何。这是一个日益活跃的研究领域，我的项目涉及高阶表示的基本问题。

The proposed projects aim to develop geometry of representations of fundamental groups of surfaces S (surface groups) into Lie groups. Finitely generate discrete groups of PSL(2, R) and PSL(2, C), called Kleinian groups, have fruitfully studied, especially, last a couple of decades. They are the groups of orientation preserving isometries of the hyperbolic spaces of dimension two and three, respectively, and represent negatively curved geometry. They play central roles in the Teichmüller theory, the study of Riemann surfaces, and 3-dimensional hyperbolic manifolds, in particular, in relation with geometrization of 3-dimensional topological manifolds. The theme of my proposed projects is to generalize interesting properties of Kleinian groups and Teichmüller theory, to more general representations and to relate geometry, topology, algebra, analysis and dynamics. One research direction of this proposal is to understand geometry of representations of the surface group into PSL(2, R) or PSL(2, C) whose images are non-discrete subgroups (non-discrete representations). Those projects seek for the geometric meaning of non-discrete representations. In particular, they are closely related to the dynamics of the action of map- ping class groups on the character varieties, the space of surface group representations. Another research direction is to study representations of surface groups to higher rank Lie groups, in particular, PSL(3, R) and PSL(3, C). While rank-one Lie groups correspond to "negatively curved geometry", higher rank Lie groups correspond to "non-positively curved geometry" and give rich geometry. It is an increasingly active area of research, and my projects concern basic problems of higher-rank representations.