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 $\Gamma$, we produce a new graph through a purely combinatorial procedure, and call it the extension graph ${\Gamma }^e$ of $\Gamma$. We produce a second graph ${\Gamma }_k^e$, the clique graph of ${\Gamma }^e$, by adding an extra vertex for each complete subgraph of ${\Gamma }^e$. We prove that each finite induced subgraph $\Lambda$ of ${\Gamma }^e$ gives rise to an inclusion $A\left(\Lambda \right)\to A\left(\Gamma \right)$. Conversely, we show that if there is an inclusion $A\left(\Lambda \right)\to A\left(\Gamma \right)$ then $\Lambda$ is an induced subgraph of ${\Gamma }_k^e$. 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 $A\left(P_4\right)$ embeds in $A\left(\Gamma \right)$ if and only if $P_4$ is an induced subgraph of $\Gamma$. We prove that if $F$ is any finite forest then $A\left(F\right)$ embeds in $A\left(P_4\right)$. We recover the first author’s result on co-contraction of graphs, and prove that if $\Gamma$ has no triangles and $A\left(\Gamma \right)$ contains a copy of $A\left(C_n\right)$ for some $n\ge 5$, then $\Gamma$ contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites’ Theorem, which asserts that if $A\left(C_4\right)$ embeds in $A\left(\Gamma \right)$ then $\Gamma$ contains an induced square. We show that whenever $\Gamma$ is triangle-free and $A\left(\Lambda \right)<A\left(\Gamma \right)$ then there is an undistorted copy of $A\left(\Lambda \right)$ in $A\left(\Gamma \right)$. Finally, we determine precisely when there is an inclusion $A\left(C_m\right)\to A\left(C_n\right)$ and show that there is no “universal” two–dimensional right-angled Artin group.