Topics in Measurable Group Theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Abstract Award: DMS-0905977 Principal Investigator: Alexander Furman The proposed project contains three research directions involving infinite groups and dynamics of group actions. These three directions include: (1) A uniform approach to a number of higher rank superrigidity phenomena. This approach, based on the notion of a generalized Weyl groups, should cover many known results, including the original Margulis' superrigidity, and its cocycle version due to Zimmer, results for abstract products, new results for groups acting on exotic affine buildings. (2) Measure-theoretic notion of imbedding between countable groups. The project is to study the resulting equivalence classes of Measurably Bi-Imbeddable groups, and the order between these classes. The techniques include cocycle superrigidity results obtained in part (1). (3) "Moduli space" of invariant metrics on a Gromov-hyperbolic group taken up to bounded distortion. The guiding and motivating source for such metrics are metrics lifted from Riemannian metrics of negative curvature on a fixed closed manifold. The project belongs to an area of research where Algebra, Geometry and Dynamics interact. The common theme of the project is the study of intrinsic symmetries of certain structures which appear in Dynamics, Algebra and Geometry, and the impact that such intrinsic symmetries have on the studied objects. The proposed problems suggest new points of view and possible generalizations to some important recent results in these areas.