Measurable bounded cohomology of measured groupoids

Abstract

We introduce the notion of measurable bounded cohomology for measured groupoids, extending continuous bounded cohomology of locally compact groups. We show that the measurable bounded cohomology of the semidirect groupoid associated to a measure class preserving action of a locally compact group $G$ on a standard Borel space is isomorphic to the continuous bounded cohomology of $G$ with twisted coefficients. We also prove the invariance of measurable bounded cohomology under similarity. As an application, we compare the bounded cohomology of (weakly) orbit equivalent actions and of measure equivalent groups. In this way we recover an isomorphism in bounded cohomology similar to one proved by Monod and Shalom. Other relevant consequences are related to the cohomological vanishing for actions of the Thompson group $F$, of higher rank lattices and of lattices in products of locally compact groups. We obtain a variant of the Eckmann-Shapiro isomorphism for transitive actions. In the case of a higher rank simple Lie group, we show that the cohomology of the action is actually determined by the usual cohomology of a suitable lattice. For amenable groupoids, we prove that the measurable bounded cohomology is trivial. This generalizes previous results by Monod, Anantharaman-Delaroche and Renault, and Blank.