Invariant percolation and measured theory of nonamenable groups

Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $\mathrm{\Gamma }$ 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 $([0, 1]^\Gamma, \lambda^\Gamma)$suchthatalmostevery$\Gamma$-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.