Invariant percolation and measured theory of nonamenable groups

Details Invariant percolation and measured theory of nonamenable groups Invariant percolation and measured theory of nonamenable groups

2011-06-27

arxiv.org/abs/1106.5337v1

Mathematics

Group Theory

Bourbaki seminar, 33 pages

Scientific paper

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

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