5 problems

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

Weighted Aldous–Lyons soficity conjecture. There exists a sequence of $K$-weighted graphs ${G_n,p_n}_{n=1}^\infty$ such that $\alpha$ is the limit of ${G_n,p_n}_{n=1}^\infty$.

Primitive-element conjecture. There exists an ergodic element $f \in [\mathcal{R}]$ such that the subequivalence relation generated by $f$ is primitive in ${\mathcal{R}}$.

Thom's conjecture. There are no discrete free subgroups of rank $2$ in the full group of the hyperfinite ergodic equivalence relation.

Let $\Gamma$ be a countable group acting ergodically and essentially freely, by measure-preserving Borel automorphisms, on a standard Borel probability space $X$. Let…