Measured group theory, percolation and non-amenability
Amenability of groups is a concept introduced by J. von Neumann in his seminal article (1929) to explain the so-called Banach-Tarski paradox. It is easily shown that the free groups F on two generators are non-amenable. It follows that the countable discrete groups containing F are non-amenable. von Neumann’s problem asked whether the converse holds true. In the 80's Ol'shansk˘ıi showed that his Tarski monsters are counter-examples. However, in order to extend certain results from groups containing F to any non-amenable countable group Γ, it may be enough to know that Γ contains F in a more dynamical sense. Namely, to know that Γ admits an ergodic probability measure preserving action on some standard space for which the orbits can be partitioned into orbits of some ergodic free action of F.
The solution to this measurable von Neumann's problem involves percolation theory on Cayley graphs and measured laminations by subgraphs. I will present an introduction to this subject.