# Groups, Geometry, and Dynamics

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**Volume 5, Issue 1, 2011, pp. 1–15**

**DOI: 10.4171/GGD/114**

Published online: 2011-01-19

Orbit equivalence, coinduced actions and free products

Lewis Bowen^{[1]}(1) The University of Texas at Austin, USA

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 $\Gamma_i = G_i *H$ 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.

*Keywords: *Coinduced, orbit equivalence, Bernoulli shifts

Bowen Lewis: Orbit equivalence, coinduced actions and free products. *Groups Geom. Dyn.* 5 (2011), 1-15. doi: 10.4171/GGD/114