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 , we produce a new graph through a purely combinatorial procedure, and call it the extension graph e of . We produce a second graph e k , the clique graph of e , by adding an extra vertex for each complete subgraph of e . We prove that each finite induced subgraph of e gives rise to an inclusion A./ ! A./. Conversely, we show that if there is an inclusion A./ ! A./ then is an induced subgraph of 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./ if and only if P 4 is an induced subgraph of . 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 has no triangles and A./ contains a copy of A.C n / for some n 5, then contains a copy of C m for some 5 m n. We also recover Kambites' Theorem, which asserts that if A.C 4 / embeds in A./ then contains an induced square. We show that whenever is triangle-free and A./ < A./ then there is an undistorted copy of A./ in A./. Finally, we determine precisely when there is an inclusion A.C m / ! A.C n / and show that there is no "universal" two-dimensional right-angled Artin group.