Orbit equivalence, coinduced actions and free products

Abstract

The following result is proven. Let \(G_1 \cc^{T_1} (X_1,\mu_1)\) and \(G_2 \cc^{T_2} (X_2,\mu_2)\) be orbit-equivalent, 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 \(\Gamma_i = G_i *H\) be the free product. Then the actions of \(\Gamma_1\) and \(\Gamma_2\) coinduced from \(T_1\) and \(T_2\) are orbit-equivalent. As an application, it is shown that if \(\Gamma\) is a free group, then all nontrivial Bernoulli shifts over \(\Gamma\) are orbit-equivalent.