5 problems

$L^2$-Hall characterization conjecture. The following conditions should be equivalent:

Graphs of free groups with cyclic edge groups. Then $\rho_+(X)\leq 0$. Furthermore, $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

Cyclic subgroup separability and virtual specialness conjecture. The following conditions on $G$ should be equivalent:

Quasi-isometric rigidity conjecture. If $\Gamma$ is the fundamental group of a graph of groups with a JSJ decomposition in which all non-cyclic vertex groups are rigid free groups,…

Let $G$ be a group that splits as a finite graph of finitely generated free groups. An algorithm is said to certify that $G$ has infinite stature with respect to its vertex groups…