Embedability between right-angled Artin groups

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 $Λ$ of $\gam^e$ gives rise to an inclusion $A(Λ)\to A(\gam)$. Conversely, we show that if there is an inclusion $A(Λ)\to A(\gam)$ then $Λ$ 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.