Invariant Percolation and Measured Theory of Nonamenable Groups

The notion of amenability was introduced in 1929 by J. von Neumann [68] in order to explain the Banach-Tarski paradox. A countable discrete group Γ is amenable if there exists a left-invariant mean φ : `∞(Γ) → C. The class of amenable groups is stable under subgroups, direct limits, quotients and the free group F2 on two generators is not amenable. Knowing whether or not the class of amenable groups coincides with the class of groups without a nonabelian free subgroup became known as von Neumann’s problem. It was solved in the negative by Ol’shanskii [50]. Adyan [1] proved that the free Burnside groups B(m,n) with m generators, of exponent n (n ≥ 665 and odd) are nonamenable. Ol’shanskii and Sapir [51] also constructed examples of finitely presented nonamenable groups without a nonabelian free subgroup.