Orbit Equivalence and Measured Group Theory

We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International Congress of Mathematicians, Hyderabad, India - International Congress of Mathematicians (ICM), Hyderabad : India (2010

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.Comment: 11 pages, title changed, a part of contents remove

Rigidity for von Neumann algebras and their invariants

We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.Comment: ICM 2010 Proceedings tex