Embeddability of right-angled Artin groups on complements of trees

Abstract

For a finite simplicial graph $Γ$, let $A(Γ)$ denote the right-angled Artin group on $Γ$. Recently Kim and Koberda introduced the extension graph $Γ^e$ for $Γ$, and established the Extension Graph Theorem: for finite simplicial graphs $Γ_1$ and $Γ_2$ if $Γ_1$ embeds into $Γ_2^e$ as an induced subgraph then $A(Γ_1)$ embeds into $A(Γ_2)$. In this article we show that the converse of this theorem does not hold for the case $Γ_1$ is the complement of a tree and for the case $Γ_2$ is the complement of a path graph.