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

A spectral sequence to compute L2-Betti numbers of groups and groupoids

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.Comment: added remark 4.9 about applying spectral sequence in a non-ergodic situation; minor correction