Stabilizers for ergodic actions of higher rank semisimple groups


Let $G$ be a connected semisimple Lie group with finite center and $\mathbb{R}$-rank $\ge 2$. Suppose that each simple factor of $G$ either has $\mathbb{R}^2$-rank $\ge 2$ or is locally isomorphic to $\mathrm{Sp}(1,n)$ or $F_{4(-20)}$. We prove that any faithful, irreducible properly ergodic, finite measure-preserving action of $G$ is essentially free. We extend the result to reducible actions and actions of lattices.



Garrett Stuck

Robert J. Zimmer