Path lifting properties and embedding between RAAGs

Abstract

For a finite simplicial graph $\Gamma$, let $G(\Gamma )$ denote the right-angled Artin group on the complement graph of $\Gamma$. In this article, we introduce the notions of "induced path lifting property" and "semi-induced path lifting property" for immersions between graphs, and obtain graph theoretical criteria for the embedability between right-angled Artin groups. We recover the result of S.-h.{} Kim and T.{} Koberda that an arbitrary $G(\Gamma )$ admits a quasi-isometric group embedding into $G(T)$ for some finite tree $T$. The upper bound on the number of vertices of $T$ is improved from $2^{2^{(m-1{)}^2}}$ to $m{2}^{m-1}$, where $m$ is the number of vertices of $\Gamma$. We also show that the upper bound on the number of vertices of $T$ is at least $2^{m/4}$. Lastly, we show that $G(C_m)$ embeds in $G(P_n)$ for $n⩾2m-2$, where $C_m$ and $P_n$ denote the cycle and path graphs on $m$ and $n$ vertices, respectively.

published version in Journal of Algebra 448 (2016) 575-594

References in corpus (3)

• Anti-trees and right-angled Artin subgroups of braid groups
• On the Subgroups of Right Angled Artin Groups and Mapping Class Groups
• Embeddings between partially commutative groups: two counterexamples

Cited by in corpus (3)

• Embeddability of right-angled Artin groups on complements of trees
• Actions of right-angled Artin groups in low dimensions
• Acylindricity of the action of right-angled Artin groups on extension graphs