Dye's reconstruction theorem implies that any isomorphism between full groups of ergodic pmp equivalence relations must come from an orbit equivalence. In this talk, motivated by Matte Bon's remarkable results on actions of topological full groups by homeomorphisms, we will investigate boolean pmp actions of full groups of ergodic pmp equivalence relations. In particular, this will allow us to understand homomorphisms between such full groups. A nice corollary of our work and of the Connes-Weiss theorem for ergodic pmp equivalence relations is the following purely full group-theoretic characterization of property (T), answering a question of Bekka: a pmp ergodic equivalence relation has (T) iff every ergodic non free boolean action of its full group is strongly ergodic.

This is joint work with Alessandro Carderi an Alice Giraud, and part of Alice's PhD thesis."