Path lifting properties and embedding between RAAGs

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.