Embedability between right-angled Artin groups

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph $\text{gam}$, we produce a new graph through a purely combinatorial procedure, and call it the extension graph ${\text{gam}}^e$ of $\text{gam}$. We produce a second graph ${\text{gam}}_k^e$, the clique graph of ${\text{gam}}^e$, by adding extra vertices for each complete subgraph of ${\text{gam}}^e$. We prove that each finite induced subgraph $\mathrm{\Lambda }$ of ${\text{gam}}^e$ gives rise to an inclusion $A(\mathrm{\Lambda })\to A(\text{gam})$. Conversely, we show that if there is an inclusion $A(\mathrm{\Lambda })\to A(\text{gam})$ then $\mathrm{\Lambda }$ is an induced subgraph of ${\text{gam}}_k^e$. 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 $A(P_4)$ embeds in $A(\text{gam})$ if and only if $P_4$ is an induced subgraph of $\text{gam}$. 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 $\text{gam}$ has no triangles and $A(\text{gam})$ contains a copy of $A(C_n)$ for some $n\ge 5$, then $\text{gam}$ contains a copy of $C_m$ for some $5\le m\le n$\mathrm{.}WealsorecoverKambite{s}^{\mathrm{\prime }}Theorem,whichassertsthatif$A(C_4)$embedsin$A(\gam)$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.