Dynamics and geometry of large groups

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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.