#### Volume 17, issue 1 (2013)

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Embedability between right-angled Artin groups

### Sang-hyun Kim and Thomas Koberda

Geometry & Topology 17 (2013) 493–530
##### 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 ${\Gamma }_{k}^{e}$, 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 $A\left(\Lambda \right)\to A\left(\Gamma \right)$. Conversely, we show that if there is an inclusion $A\left(\Lambda \right)\to A\left(\Gamma \right)$ then $\Lambda$ is an induced subgraph of ${\Gamma }_{k}^{e}$. 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 $A\left({P}_{4}\right)$ embeds in $A\left(\Gamma \right)$ if and only if ${P}_{4}$ is an induced subgraph of $\Gamma$. We prove that if $F$ is any finite forest then $A\left(F\right)$ embeds in $A\left({P}_{4}\right)$. We recover the first author’s result on co-contraction of graphs, and prove that if $\Gamma$ has no triangles and $A\left(\Gamma \right)$ contains a copy of $A\left({C}_{n}\right)$ for some $n\ge 5$, then $\Gamma$ contains a copy of ${C}_{m}$ for some $5\le m\le n$. We also recover Kambites’ Theorem, which asserts that if $A\left({C}_{4}\right)$ embeds in $A\left(\Gamma \right)$ then $\Gamma$ contains an induced square. We show that whenever $\Gamma$ is triangle-free and $A\left(\Lambda \right) then there is an undistorted copy of $A\left(\Lambda \right)$ in $A\left(\Gamma \right)$. Finally, we determine precisely when there is an inclusion $A\left({C}_{m}\right)\to A\left({C}_{n}\right)$ and show that there is no “universal” two–dimensional right-angled Artin group.

##### Keywords
right-angled Artin group, mapping class group, surface group, co-contraction
Primary: 20F36
##### Publication 