Orbit equivalence, coinduced actions and free products

Abstract

The following result is proven. Let $G_1{\curvearrowright }^{T_1}(X_1,{\mu }_1)$ and $G_2{\curvearrowright }^{T_2}(X_2,{\mu }_2)$ be orbit equivalent (OE), essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let ${\Gamma }_i=G_i\ast H$ be the free product. Then the actions of ${\Gamma }_1$ and ${\Gamma }_2$ coinduced from $T_1$ and $T_2$ are OE. As an application, it is shown that if $\Gamma$ is a free group, then all nontrivial Bernoulli shifts over $\Gamma$ are OE.