Appears in collection : Geometry in non-positive curvature and Kähler groups

Measure equivalence is a measurable analogue of quasi-isometry. For instance, two lattices (co-compact or not) in a same Lie group are measurably equivalent by definition. We prove that for N bigger or equal than 3, any countable group that is measure equivalent to Out(Fn) is virtually isomorphic to it. I will discuss some of the tools introduced for this proof, and in particular, a notion of a canonical dynamic decomposition associated to a subgroup of which somehow generalizes the dynamical decomposition of a surface associated to a subgroup of Out(Fn) the mapping class group. This is a joint work with Camille Horbez.