Embeddability of right-angled Artin groups on the complements of linear forests

TL;DR

This paper establishes a graph-theoretic criterion for embedding right-angled Artin groups based on the structure of their defining graphs and extension graphs, simplifying the understanding of group embeddings.

Contribution

It provides a reduction of embedding problems of right-angled Artin groups on complements of linear forests to graph embedding problems.

Findings

Embedding of right-angled Artin groups corresponds to full graph embeddings.

Reduction to extension graph embeddings simplifies the analysis.

Results apply specifically to complements of linear forests.

Abstract

In this article, we prove that embeddings of right-angled Artin group $A_1$ on the complement of a linear forest into another right-angled Artin group $A_2$ can be reduced to full embeddings of the defining graph of $A_1$ into the extension graph of the defining graph of $A_2$.