Topics in Measurable Group Theory

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AbstractAward: DMS-0905977Principal Investigator: Alexander FurmanThe proposed project contains three research directions involvinginfinite groups and dynamics of group actions. These threedirections include: (1) A uniform approach to a number of higherrank superrigidity phenomena. This approach, based on the notionof a generalized Weyl groups, should cover many known results,including the original Margulis' superrigidity, and its cocycleversion due to Zimmer, results for abstract products, new resultsfor groups acting on exotic affine buildings. (2)Measure-theoretic notion of imbedding between countable groups.The project is to study the resulting equivalence classes ofMeasurably Bi-Imbeddable groups, and the order between theseclasses. The techniques include cocycle superrigidity resultsobtained in part (1). (3) "Moduli space" of invariant metrics ona Gromov-hyperbolic group taken up to bounded distortion. Theguiding and motivating source for such metrics are metrics liftedfrom Riemannian metrics of negative curvature on a fixed closedmanifold.The project belongs to an area of research where Algebra,Geometry and Dynamics interact. The common theme of the projectis the study of intrinsic symmetries of certain structures whichappear in Dynamics, Algebra and Geometry, and the impact thatsuch intrinsic symmetries have on the studied objects. Theproposed problems suggest new points of view and possiblegeneralizations to some important recent results in these areas.