Invariant percolation and measured theory of nonamenable groups

Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $Γ$ contains measurably the free group $\mathbf F_2$ on two generators: there exists a probability measure-preserving, essentially free, ergodic action of $\mathbf F_2$ on $([0, 1]^Γ, λ^Γ)$ such that almost every $Γ$-orbit of the Bernoulli shift splits into $\mathbf F_2$-orbits. A combination of this result and works of Ioana and Epstein shows that every countable, discrete, nonamenable group admits uncountably many non-orbit equivalent actions.

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