7 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…

Let $G$ be a finitely generated group containing a copy of the free group $\mathbb{F}_d$ with finite index, where $d\geq 2$, and let $\mu$ be a symmetric, adapted random walk on…

Let $F_n$ be the free group of rank $n$. An invariant random subgroup (IRS) is co-sofic if it is a weak- limit in $\operatorname{IRS}(G)$ of IRSs supported on lattices. Aldous--Lyo…

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

Existence question. Does every non-discrete locally compact group admit a non-trivial IRS?

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…