Orbit equivalence, coinduced actions and free products

TL;DR

The paper proves that coinduced actions from orbit-equivalent free actions of countable groups remain orbit-equivalent when extended via free products, and applies this to show all Bernoulli shifts over free groups are orbit-equivalent.

Contribution

It establishes that coinduction preserves orbit equivalence in free product extensions and demonstrates the orbit-equivalence of all Bernoulli shifts over free groups.

Findings

Coinduced actions from orbit-equivalent actions are orbit-equivalent after free product extension.

All nontrivial Bernoulli shifts over free groups are orbit-equivalent.

The result applies to a broad class of group actions, linking orbit equivalence and free product constructions.

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.