Embeddability of right-angled Artin groups on complements of trees

Abstract

For a finite simplicial graph $\Gamma$, let $A(\Gamma )$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph ${\Gamma }^e$ for $\Gamma$, and established the Extension Graph Theorem: for finite simplicial graphs ${\Gamma }_1$ and ${\Gamma }_2$ if ${\Gamma }_1$ embeds into ${\Gamma }_2^e$ as an induced subgraph then $A({\Gamma }_1)$ embeds into $A({\Gamma }_2)$. In this article we show that the converse of this theorem does not hold for the case ${\Gamma }_1$ is the complement of a tree and for the case ${\Gamma }_2$ is the complement of a path graph.

published version in International Journal of Algebra and Computation

References in corpus (3)

• Anti-trees and right-angled Artin subgroups of braid groups
• Embeddability and universal theory of partially commutative groups
• Path lifting properties and embedding between RAAGs

Cited by in corpus (1)

• Acylindricity of the action of right-angled Artin groups on extension graphs