Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions (Q391962)

In this paper the authors give another proof for \textit{L. Bowen}'s result [Groups Geom. Dyn. 5, No. 1, 17--38 (2011; Zbl 1259.37002)] saying that if \(1<n,m<\infty\) then all Bernoulli-shift actions over \(\mathbb{F}_n\) and \(\mathbb{F}_m\) are stably orbit equivalent, where \(\mathbb{F}_n\) denotes the free group of rank \(n\) .NEWLINENEWLINEThe difference between the two approaches is the following. In his original paper, Bowen proved that every Bernoulli-shift actions over a free group \(\mathbb{F}_n\) of rank \(n\) \((1<n<\infty)\) is stable orbit equivalent to a Bernoulli-shift action over \(\mathbb{F}_2,\) and then he applied his result that all Bernoulli-shift actions over a free group are orbit equivalent [\textit{L. Bowen}, Groups Geom. Dyn. 5, No. 1, 1--15 (2011; Zbl 1257.37004)]. On the other hand, in this paper the authors prove the results of Bowen too, but they use a different method, still using Bowen's general idea. First, they notice that the co-induction of a Bernoulli action is again a Bernoulli action over the same base space. Then, they prove that the co-induced actions to \(\mathbb{F}_n\) are orbit equivalent and so they conclude that, for fixed \(n\) and varying base probability space, the Bernoulli actions over \(\mathbb{F}_n\) are orbit equivalent. As last step in their ideas, they prove that all Bernoulli actions of \(\mathbb{F}_2\) are stably orbit equivalent with all Bernoulli actions of \(\mathbb{F}_m,\) \(m\geq 2,\) and by transitivity of stable orbit equivalence, that all Bernoulli actions of \(\mathbb{F}_n\) and \(\mathbb{F}_m\) are stably orbit equivalent.