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

TL;DR

This paper proves that Bernoulli actions of free groups are stably orbit equivalent regardless of base spaces and shows certain factor actions are isomorphic to Bernoulli actions, advancing understanding of their structural properties.

Contribution

It provides an elementary proof of stable orbit equivalence for Bernoulli actions of free groups and establishes isomorphism of specific factor actions with Bernoulli actions.

Findings

Bernoulli actions of free groups are stably orbit equivalent.

Certain factor actions are isomorphic to Bernoulli actions.

The methods simplify previous proofs and extend known results.

Abstract

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 n infinite amenable groups, the factor K^{\Gamma}/K of the Bernoulli action of \Gamma on K^{\Gamma} by the diagonal action of K, is isomorphic with a Bernoulli action of \Gamma.