Ideal bicombings for hyperbolic groups and applications

Abstract. For every hyperbolic group, we construct an ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [26] hold for all non-elementary hyperbolic groups and their non-elementary subgroups. For any subgroup Γ of a hyperbolic group, this invariant provides a class in H 2 b (Γ, ℓ2 (Γ)) which vanishes if and only if Γ is elementary. As an additional application, we derive a superrigidity result for homomorphisms of general irreducible lattices to hyperbolic groups. 1.