Dynamics and geometry of large groups

该提议涉及遍历理论与无限群研究之间的相互作用。这包括使用遍历的理论概念，特别是修改后的群体边界概念，来研究各种群体的超刚性现象。在另一个方向上，建议使用嵌入在某些经典动力系统中的群作用来证明Lyapunov谱的简单性--这是系统混沌行为的一个重要的定量特征。还提出了“测量群论”中的其他问题。拟议的项目将继续探索代数、几何和动力学之间的许多令人惊讶的联系。该建议的一部分描述了某些动力系统的某些隐式对称性产生群的刚性的方法。在另一个方向上，具有某些性质的群的存在被猜想为产生某些动力学性质。

The proposal concerns interactions between ergodic theory and the study of infinite groups. This includes the use of ergodic theoretic concepts, in particular a revised notion of group boundaries, for the study of superrigidity phenomena for various groups. In another direction, it is suggested to use group actions imbedding in certain classical dynamical systems to prove simplicity of the Lyapunov spectrum - an important quantitative characteristic of chaotic behavior of a system. Additional problems in "measured group theory" are proposed.The proposed project will continue to explore many of the surprising connections between Algebra, Geometry and Dynamics. Part of the proposal describes ways in which some implicit symmetries of certain dynamical systems yield rigidity of groups. In the other direction, presence of groups with certain properties, is conjectured to yield certain dynamical properties.