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

TL;DR

This paper characterizes when groups formed as graphs of free groups with cyclic edge groups are residually finite, linking this property to the residual finiteness of Baumslag-Solitar subgroups and providing a graph-based criterion.

Contribution

It proves a conjecture of Wise by providing a graph-theoretic characterization of residual finiteness for these groups.

Findings

Residual finiteness is equivalent to all Baumslag-Solitar subgroups being residually finite.

A finite labeled graph can be constructed from a presentation to determine residual finiteness.

The characterization simplifies checking residual finiteness in these 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.