## Geometry & Topology

### 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 $Γ$, we produce a new graph through a purely combinatorial procedure, and call it the extension graph $Γe$ of $Γ$. We produce a second graph $Γke$, the clique graph of $Γe$, by adding an extra vertex for each complete subgraph of $Γe$. We prove that each finite induced subgraph $Λ$ of $Γe$ 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 $Γke$. 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. We show that whenever $Γ$ is triangle-free and $A(Λ) then there is an undistorted copy of $A(Λ)$ in $A(Γ)$. 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.

#### Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 493-530.

Dates
Accepted: 20 November 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732530

Digital Object Identifier
doi:10.2140/gt.2013.17.493

Mathematical Reviews number (MathSciNet)
MR3039768

Zentralblatt MATH identifier
1278.20049

Subjects
Primary: 20F36: Braid groups; Artin groups

#### Citation

Kim, Sang-hyun; Koberda, Thomas. Embedability between right-angled Artin groups. Geom. Topol. 17 (2013), no. 1, 493--530. doi:10.2140/gt.2013.17.493. https://projecteuclid.org/euclid.gt/1513732530

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