the subject of this talk is at the crossroads of functional analysis, ergodic theory and group theory. using a construction by murray and von neumann (1943), ergodic actions of countable groups on probability spaces give rise to algebras of operators on a hilbert space, called von neumann algebras. in a joint work with sorin popa, we proved that such crossed product von neumann algebras by free groups or, more generally, by hyperbolic groups have a unique cartan subalgebra. i will explain this result and its consequences for the classification of crossed products by free groups.