Résumé : 

The Poisson-Furstenberg boundary is an object associated to a pair of a group and a measure on said group that captures the asymptotic properties of the random walk. 

The Furstenberg boundary is another, topological object associated with a group, that turned out to be deeply connected with C*-simplicity and the unique trace property. Motivated by the desire to extend this connection, I will present new examples where the group acts non-freely on its Poisson-Furstenberg boundary.