文摘
For a finite simplicial graph Γ, let 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) 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 34b2f186807eea46e7d" title="Click to view the MathML source">22(m−1)2 to m2m−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/4. Lastly, we show that b4001ab6617791908774954" title="Click to view the MathML source">G(Cm) embeds in G(Pn) for n⩾2m−2, where Cm and Pn denote the cycle and path graphs on m and n vertices, respectively.