Conjecture on cyclic subgroup separability and virtual compact specialness for graphs of groups

Let $G$ be a finitely generated graph of groups with free abelian vertex groups and cyclic edge groups. A group is cyclic subgroup separable if every cyclic subgroup can be separated from each element outside it by a finite quotient. An edge group is isolated when it has the isolation property used in the source; a group is virtually compact special when it has a finite-index subgroup that is the fundamental group of a compact special cube complex.

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

• $G$ is cyclic subgroup separable;
• $G$ has a finite-index subgroup with isolated edge groups;
• $G$ is virtually compact special.

Furthermore, if $G$ is freely indecomposable, then $G$ is cyclic subgroup separable if and only if it is residually finite and non-solvable.

The conjecture proposes algebraic, decomposition-theoretic, and geometric characterizations of the relevant graphs of groups. The paper presents it as a belief about the class, and attributes the terminology “virtually primitive” for the isolated-edge condition to Hoda; no resolution is supplied here.