7 problems

Let $\Gamma$ be a countable group. An essentially free, ergodic, probability measure-preserving action of $\Gamma$ on a standard probability space has an associated cost, and…

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 $(X,bcmu)$ be a standard probability space, and let $\mathcal{G}$ be a locally finite Borel graph on $X$. Say that $\mathcal{G}$ is $\mu$-nowhere smooth if no positive-measure…

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 $\mathcal{R}$ be a discrete, measure-preserving equivalence relation, let $c(\mathcal{R})$ denote its cost, and let $\beta^{(2)}_{1}(\mathcal{R})$ denote its first $l^{2}$-Bett…