However, from the dynamical perspective, knowing that every measure-preserving action is free is not a complete answer: since lattices are nonamenable they have many natural actions which do not admit invariant measures. The natural question to ask is whether or not every minimal action of \Gamma on a compact metric space X admits a nonsingular measure so that the action is essentially free.
I answer this in the affirmative. Let \mu be the measure on \Gamma so that the Poisson boundary of (\Gamma,\mu) is that of G. We show that every action of \Gamma on a nonatomic probability space (X,\nu) where \nu is \mu-stationary is essentially free (such measures exist on every compact metric space where \Gamma acts).
I will also present some conjectures about the non-commutative generalization of this statement involving an apparently new notion of a ``stationary representation" of a lattice on a II_1 factor or Hilbert space that suggests that the rigidity phenomena of lattices is even stronger than is currently known.