The Stuck--Zimmer conjecture

Let $G$ be a semisimple Lie group of rank at least $2$, and let a p.m.p. action of $G$ be irreducible, meaning that every factor of $G$ acts ergodically. An action is essentially free when almost every point has trivial stabilizer, and transitive when it has a single orbit up to null sets. The Stuck--Zimmer conjecture. Every irreducible p.m.p. action of $G$ is essentially free or transitive, without assuming that $G$ has Kazhdan's property $(T)$. The theorem is known when at least one factor of $G$ has property $(T)$, as confirmed by Hartman and Tamuz, but remains open in general; the first case mentioned in the source is $G=\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$.