4 problems

Connectedness conjecture. There exists a number $\delta$ with $\frac{1}{2}<\delta<1$ such that $\mathcal{E}$ is connected for every $p\geqslant\delta$.

For each $n\geq 1$, let $\mathcal{BD}^*_n$ be the class of Bratteli diagrams of exact rank $n$, and let $BD^*_n$ be the equivalence relation induced by equivalence of Bratteli diag…

Mixing conjecture. The vertical flow on $S(\mathcal{B},w^\pm, \leq_{r,s})$ is mixing.

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…