Path lifting properties and embedding between RAAGs

Original · EN

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 embedability 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 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 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.