4 problems

Let $\Gamma$ be a center-free irreducible higher-rank lattice. Stuck–Zimmer's IRS rigidity conjecture. For every ergodic probability-measure-preserving action of $\Gamma$ on a prob…

Suppose $\Gamma\overset{\alpha}{\curvearrowright}(X,\mu)$ is a probability-measure-preserving action, with associated invariant random subgroup $H_\alpha$ and orbit equivalence rel…

Weak uniform discreteness conjecture. The space $\operatorname{DIRS}(G)$ is weakly uniformly discrete. The statement extends weak uniform discreteness beyond the no-discrete-small-…

Let $G$ be the Lie group under consideration, let $k$ be a degree, and let an IRS $\mu$ of $G$ be $\varepsilon$-discrete in the sense used in the source. Let $\nu_\mu^k$ be the spe…