Key points are not available for this paper at this time.
Every stationary action of a strongly irreducible lattice or commensurator of such a lattice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely. Consequently, every minimal action of such a lattice on an infinite compact metric space is topologically free.
Darren Creutz (Mon,) studied this question.