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(e)(k), 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(Lambda) -> A(Gamma). Conversely, we show that if there is an inclusion A(Lambda) -> A(Gamma) then Lambda is an induced subgraph of Gamma(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(Gamma) if and only if P-4 is an induced subgraph of Gamma. 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 Gamma has no triangles and A(Gamma) contains a copy of A(C-n) for some n >= 5, then Gamma 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(Gamma) then Gamma contains an induced square. We show that whenever Gamma is triangle-free and A(Lambda) < A(Gamma) then there is an undistorted copy of A(Lambda) in A(Gamma). 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.