Invariant percolation and measured theory of nonamenable groups

Abstract

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.

Bourbaki seminar, 33 pages

References in corpus (2)

• Minimal spanning forests
• Orbit inequivalent actions of non-amenable groups

Cited by in corpus (1)

• Solidity of type III Bernoulli crossed products