Path lifting properties and embedding between RAAGs

TL;DR

This paper introduces new graph-theoretic criteria for embedding right-angled Artin groups, improving bounds on the size of trees into which these groups can embed, and explores embeddings between specific RAAGs based on graph properties.

Contribution

It develops the notions of induced and semi-induced path lifting properties, providing criteria for RAAG embeddings and improving bounds on the size of trees for embeddings.

Findings

Improved upper bound on the number of vertices in trees embedding RAAGs from double exponential to linear exponential.

Established that $G(C_m)$ embeds into $G(P_n)$ for $n extgreater= 2m-2$.

Revealed new graph-theoretic conditions for RAAG embeddability based on path lifting properties.

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 $m2^{m-1}$, 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 $n\geqslant 2m-2$, where $C_m$ and $P_n$ denote the cycle and path graphs on $m$ and $n$ vertices, respectively.