Amenability for groups is a concept introduced by J. von Neumann in his seminal article (1929) tin connection with the so-called Banach-Tarski paradox. It is easily shown that the free group F on two generators is non-amenable. It follows that the countable discrete groups containing F are non-amenable. von Neumann's problem examines whether the converse holds true. In the 80's Ol'shanskii showed that his Tarski monsters lead to counter-examples. However, in order to extend certain results from groups containing F to any non-amenable countable group G, it may be enough to know that G contains F in a more dynamical sense. Namely, it may be sufficient to find a probability measure preserving free action of G whose orbits contain the orbits of some free action of F.

The solution to this "measurable von Neumann's problem" involves percolation theory on Cayley graphs, measured laminations by subgraphs and some invariants of dynamical systems. I will present an introduction to this subject, with some examples, pictures, and movies. We may even take this opportunity to start a ping-pong game...