Right-angled Artin groups and full subgraphs of graphs

TL;DR

This paper characterizes when right-angled Artin groups defined by complements of graphs can be embedded into each other, establishing a precise link with full subgraph embeddings for linear forests and highlighting exceptions for non-linear forests.

Contribution

It proves a complete characterization of embeddings of right-angled Artin groups via full subgraph relations for linear forests, and shows the limitations of this correspondence for non-linear forests.

Findings

Embedding of G(Λ) into G(Γ) for linear forests Λ corresponds exactly to Λ being a full subgraph of Γ.

For non-linear forests Λ, G(Λ) can embed into some G(Γ) without Λ being a full subgraph of Γ.

The established correspondence between group embeddings and subgraph embeddings is specific to linear forests.

Abstract

For a finite graph $\Gamma$, let $G(\Gamma)$ be the right-angled Artin group defined by the complement graph of $\Gamma$. We show that, for any linear forest $\Lambda$ and any finite graph $\Gamma$, $G(\Lambda)$ can be embedded into $G(\Gamma)$ if and only if $\Lambda$ can be realised as a full subgraph of $\Gamma$. We also prove that if we drop the assumption that $\Lambda$ is a linear forest, then the above assertion does not hold, namely, for any finite graph $\Lambda$, which is not a linear forest, there exists a finite graph $\Gamma$ such that $G(\Lambda)$ can be embedded into $G(\Gamma)$, though $\Lambda$ cannot be embedded into $\Gamma$ as a full subgraph.