2026/07/24 by Jos{é} Galvez Mateos, Luis Paris
#math.GR
Let Γ be a Coxeter graph, and let S be its vertex set. An elemental complex over Γ is an abstract simplicial complex \mathcal D on S that contains every pair \s, t\ with s ≠ t and ms,t ≠ ∞. To each elemental complex \mathcal D, we associate a cube complex Ξ_\mathcal D on which the virtual Artin group \rm VA [Γ] acts. We prove that Ξ_\mathcal D is connected, simply connected, and locally regular; that the action of \rm VA[Γ] on Ξ_\mathcal D is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that ΞÆ’_\mathcal D is CAT(0) if and only if \mathcal D is a flag complex.