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 K and every free product Gamma of infinite amenable groups, the factor Gamma curved right arrow K-Gamma /K of the Bernoulli action Gamma curved right arrow K-Gamma by the diagonal K-action is isomorphic with a Bernoulli action of Gamma. (C) 2012 Elsevier GmbH. All rights reserved.