A cube complex for virtual Artin groups

Let $\Gamma$ be a Coxeter graph, and let $S$ be its vertex set. An elemental complex over $\Gamma$ is an abstract simplicial complex $\mathcal D$ on $S$ that contains every pair $\{s, t\}$ with $s \neq t$ and $m_{s,t} \neq \infty$. To each elemental complex $\mathcal D$, we associate a cube complex $\Xi_\mathcal D$ on which the virtual Artin group $\rm VA [\Gamma]$ acts. We prove that $\Xi_\mathcal D$ is connected, simply connected, and locally regular; that the action of $\rm VA[\Gamma]$ on $\Xi_\mathcal D$ is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that $\XiÆ’_\mathcal D$ is CAT(0) if and only if $\mathcal D$ is a flag complex.