On residual finiteness of graphs of free groups with cyclic edge groups

Abstract

We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group $G$ is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a presentation of $G$, we construct a finite labeled graph $\Gamma$, and show that residual finiteness of $G$ is equivalent to an easily-detectable property of this graph. This characterization proves a conjecture of Wise.