6 problems

Extension conjecture. The main results of the paper should hold for full groups of minimal almost finite systems.

Let $\mathcal F:\mathbb R\curvearrowright(X,\mu)$ be a measure-preserving flow, and let $n$ be the number of its ergodic components. Write $[\mathcal F]_1$ for its $\mathrm{L}^{1}$…

Maximality conjecture. The $\mathrm L^1$ metric on the $\mathrm L^1$ full group of $\Phi$ is maximal.

AH conjecture. There exists an exact sequence

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

Let $G$ be the full group (not necessarily simple) of a Bratteli diagram, and let $D(G)$ be its commutator subgroup. Write $H=G/D(G)$, let $[g]$ denote the image of $g$ in $H$, and…