Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions

We give an elementary proof for Lewis Bowen’s theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups KK and every free product ΓΓ of infinite amenable groups, the factor Γ↷KΓ/KΓ↷KΓ/K of the Bernoulli action Γ↷KΓΓ↷KΓ by the diagonal KK-action is isomorphic with a Bernoulli action of ΓΓ.