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 $\Lambda$ 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.Comment: 35 pages. Added an appendix and a proof that the extension graph is quasi-isometric to a tre