Path lifting properties and embedding between RAAGs

For a finite simplicial graph Γ, let G(Γ)G(Γ) denote the right-angled Artin group on the complement graph of Γ. 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 embeddability between right-angled Artin groups. We recover the result of S.-h. Kim and T. Koberda that an arbitrary G(Γ)G(Γ) admits a quasi-isometric group embedding into G(T)G(T) for some finite tree T. The upper bound on the number of vertices of T   is improved from 22(m−1)222(m−1)2 to m2m−1m2m−1, where m is the number of vertices of Γ. We also show that the upper bound on the number of vertices of T   is at least 2m/42m/4. Lastly, we show that G(Cm)G(Cm) embeds in G(Pn)G(Pn) for n⩾2m−2n⩾2m−2, where CmCm and PnPn denote the cycle and path graphs on m and n vertices, respectively.