A cube complex for virtual Artin groups

Abstract

Let $\hat{I}“$ be a Coxeter graph, and let $S$ be its vertex set. An elemental complex over $\hat{I}“$ is an abstract simplicial complex $\mathcal{D}$ on $S$ that contains every pair $\{s,t\}$ with $s\ne t$ and $m_{s,t}\ne \infty$. To each elemental complex $\mathcal{D}$, we associate a cube complex $\hat{I}{ž}_{\mathcal{D}}$ on which the virtual Artin group $\mathrm{VA}[\hat{\mathrm{I}}“]$ acts. We prove that $\hat{I}{ž}_{\mathcal{D}}$ is connected, simply connected, and locally regular; that the action of $\mathrm{VA}[\hat{\mathrm{I}}“]$ on $\hat{I}{ž}_{\mathcal{D}}$ is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that $\Xi \tilde{A}†\hat{a}€{™}_{\mathcal{D}}$ is CAT(0) if and only if $\mathcal{D}$ is a flag complex.