The $L^2$-Hall characterization for graphs of free groups with cyclic edge groups

Let $X$ be a finite graph of groups with free vertex groups and cyclic edge groups, and let $G=\pi_1(X)$. Let $\Phi_X$ denote the associated graph used to identify the generalized Baumslag--Solitar components. A component is solvable when its associated generalized Baumslag--Solitar group is solvable.

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

1. $G$ has the $L^2$-Hall property.
2. Every component of $\Phi_X$ is solvable.
3. The generalized Baumslag--Solitar groups associated to the components of $\Phi_X$ all have the $L^2$-Hall property.

The paper motivates this conjecture by analogy with known criteria for subgroup separability and residual finiteness. The theorem immediately preceding it proves the $L^2$-Hall property under the stronger balanced-and-solvable, equivalently relatively hyperbolic, hypothesis; the equivalence proposed here remains open in the supplied text.