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 Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ of Γ. We produce a second graph Γ k , the clique graph of Γ, by adding extra vertices for each complete subgraph of Γ. We prove that each finite induced subgraph Λ of Γ gives rise to an inclusion A(Λ) → A(Γ). Conversely, we show that if there is an inclusion A(Λ) → A(Γ) then Λ is an induced subgraph of Γ k . These results have a number of corollaries. Let P4 denote the path on four vertices and let Cn denote the cycle of length n. We prove that A(P4) embeds in A(Γ) if and only if P4 is an induced subgraph of Γ. We prove that if F is any finite forest then A(F ) embeds in A(P4). We recover the first author’s result on co–contraction of graphs, and prove that if Γ has no triangles and A(Γ) contains a copy of A(Cn) for some n ≥ 5, then Γ contains a copy of Cm for some 5 ≤ m ≤ n. We also recover Kambites’ Theorem, which asserts that if A(C4) embeds in A(Γ) then Γ contains an induced square. Finally, we determine precisely when there is an inclusion A(Cm) → A(Cn) and show that there is no “universal” two–dimensional right-angled Artin group.