Geometric structures and surface group representations into Lie groups

拟议的项目旨在将曲面的基本群S(曲面群)的表示的几何发展成李群。有限生成的PSL(2，R)和PSL(2，C)离散群，称为Klein群，在过去的几十年里已经有了丰硕的研究成果。它们分别是二维双曲空间和三维双曲空间的保定向等距线群，表示负曲几何。它们在Teichmüler理论、Riemann曲面和三维双曲流形的研究中起着核心作用，特别是与三维拓扑流形的几何化有关。我提出的项目的主题是将克莱因群和泰希米勒理论的有趣性质推广到更一般的表示法，并将几何学、拓扑学、代数、分析和动力学联系起来。这一建议的一个研究方向是将曲面群的表示理解为PSL(2，R)或PSL(2，C)，其像是非离散子群(非离散表示)。这些项目寻求非离散表示的几何意义。特别是，它们与映射类群对字符种类、表面群表示空间的作用的动力学密切相关。另一个研究方向是研究表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，我的项目涉及更高级别表示的基本问题。

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.