Commensurating actions of groups of birational transformations

A commensurating action is an action of a group on a set along with a subset that is commensurated, in the sense that it has finite symmetric difference with each of its translates. There are natural ways to go from commensurating actions to actions on CAT(0) cube complexes and vice versa. For each variety X, we construct a natural commensurating action of the group of birational transformations of X. The commensurated subset turns out to be the set of irreducible hypersurfaces in X. Under the assumption that the acting group has Property FW (which means that every action on a CAT(0) cube complex has a fixed point), we deduce restrictions on its birational actions.   (Joint work with Serge Cantat)