群作用与非正曲率空间的几何拓扑

本项目研究自由群自同构群、矩阵群、曲面映射类群在非正曲率空间上的等距以及同胚作用的刚性问题。对于CAT(0)空间，我们考虑其上这类典型无限群的等距作用，计划证明当空间维数低于群的尺度时，这些无限群作用一定具有全局不动点。对于非正曲率流形，我们考虑其上的这类典型无限群同胚作用，计划证明当流形维数低于矩阵群尺度时，这些无限群作用一定只能是个有限群作用。此类问题与几何群论中群作用刚性公开问题息息相关，具有广泛兴趣，意义重大。一方面群的性质可以通过群作用充分反映，另一方面空间的对称性通过群作用充分表达。

In this project, we study the rigidity of actions of automorphism group of free groups, matrix groups and mapping class groups on non-positively curved spaces. For CAT(0) spaces, we consider the isometric actions and want to prove that when the dimension of the space is smaller than the sizes of groups, the group action must have a global fixed point. For non-positively curved manifolds, we consider the topological actions and want to prove that when the dimension of manifolds is smaller that the sizes of groups, the group action factors through a finite group. These are important rigidity phenomena in geometric group theory， close to related to some open problems. On one hand, the properties of groups could be studied from the actions. On the other hand, the symmetries of spaces can also be reflected through the actions.