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

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
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
Received: 20 March 2012
Accepted: 20 November 2012
Published: 2 April 2013
Proposed: Benson Farb
Seconded: Cameron Gordon, Danny Calegari
##### Authors
 Sang-hyun Kim Department of Mathematical Sciences KAIST Daejeon 305-701 Republic of Korea http://shkim.kaist.ac.kr/main/Main.html Thomas Koberda Department of Mathematics Yale University P.O. Box 208283 New Haven, CT 06511 USA 