Embedability between right-angled Artin groups

TL;DR

This paper characterizes when right-angled Artin groups embed into each other using combinatorial graph constructions, revealing new structural insights and limitations about their subgroup relationships.

Contribution

It introduces the extension graph and clique graph concepts to precisely characterize subgroup embeddings of right-angled Artin groups.

Findings

Embedding of $A(P_4)$ corresponds to $P_4$ being an induced subgraph.

Any finite forest embeds into $A(P_4)$.

No universal two-dimensional right-angled Artin group exists.

Abstract

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph $\gam$, we produce a new graph through a purely combinatorial procedure, and call it the extension graph $\gam^e$ of $\gam$. We produce a second graph $\gam^e_k$, the clique graph of $\gam^e$, by adding extra vertices for each complete subgraph of $\gam^e$. We prove that each finite induced subgraph $\Lambda$ of $\gam^e$ gives rise to an inclusion $A(\Lambda)\to A(\gam)$. Conversely, we show that if there is an inclusion $A(\Lambda)\to A(\gam)$ then $\Lambda$ is an induced subgraph of $\gam^e_k$. These results have a number of corollaries. Let $P_4$ denote the path on four vertices and let $C_n$ denote the cycle of length $n$. We prove that $A(P_4)$ embeds in $A(\gam)$ if and only if $P_4$ is an induced subgraph of $\gam$. We prove that if $F$ is any finite forest then $A(F)$ embeds in $A(P_4)$. We recover the first author's result on co--contraction of graphs and prove that if $\gam$ has no triangles and $A(\gam)$ contains a copy of $A(C_n)$ for some $n\geq 5$, then $\gam$ contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites' Theorem, which asserts that if $A(C_4)$ embeds in $A(\gam)$ then $\gam$ contains an induced square. Finally, we determine precisely when there is an inclusion $A(C_m)\to A(C_n)$ and show that there is no "universal" two--dimensional right-angled Artin group.