Measured group theory, percolation and non-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