Right-angled Artin subgroups and free products in one-relator groups

TL;DR

This paper explores conditions under which one-relator groups contain right-angled Artin subgroups, providing criteria related to graph structures and subgroup embeddings, with implications for properties like $P_{nai}$ and $C^*$-simplicity.

Contribution

It establishes new criteria for embedding right-angled Artin groups into one-relator groups based on graph properties and subgroup embeddings.

Findings

Embedding of positive submonoids implies embedding of entire Artin groups.

Characterizations of one-relator groups with property $P_{nai}$.

Criteria for $C^*$-simplicity in one-relator groups.

Abstract

We investigate criteria ensuring that a one-relator group $G$ contains a right-angled Artin subgroup $A(\Gamma)$, corresponding to a finite graph $\Gamma$. In particular, we prove that if $\Gamma$ is a forest with at least one edge and the positive submonoid $T(\Gamma)$, of $A(\Gamma)$, embeds into $G$ then so does all of $A(\Gamma)$. As by-products of our methods we obtain characterisations of one-relator groups that have property $P_{nai}$ and that are $C^*$-simple.