A note on right-angled Artin subgroups of one-relator groups

TL;DR

This paper provides a simple proof that right-angled Artin groups embedding into one-relator groups must have their defining graph as a finite forest, using elementary Bass--Serre theory.

Contribution

It offers a concise, elementary proof of a known result about the structure of graphs defining Artin groups embedded in one-relator groups.

Findings

If $A( ext{ extbackslash Gamma})$ embeds into a one-relator group, then $ ext{ extbackslash Gamma}$ is a finite forest.

The proof relies solely on elementary Bass--Serre theory and classical properties.

The result clarifies the structural limitations of right-angled Artin groups within one-relator groups.

Abstract

We give a short proof of the following result due to Howie: if $A(\Gamma)$ is a right-angled Artin group embedding into some one-relator group, then $\Gamma$ is a finite forest. The proof only uses elementary Bass--Serre theory and classical properties of one-relator groups.