Spaces Related to Virtual Artin Groups

Abstract

This work explores the topological properties of virtual Artin groups, a recent extension of the ``virtual" concept - initially developed for braids - to all Artin groups, as introduced by Bellingeri, Paris, and Thiel. For any given Coxeter graph $Γ$, we define a CW-complex $Ω(Γ)$ whose fundamental group is isomorphic to the pure virtual Artin group $\mathrm{PVA}[Γ]$, which coincides with the pure virtual braid group when $Γ$ is $A_{n-1}$. This construction generalizes the previously studied BEER complex, originally defined for pure virtual braids, to all Coxeter graphs. We investigate the asphericity of $Ω(Γ)$ and demonstrate that it holds when $Γ$ is of spherical type or of affine type, thereby characterizing $Ω(Γ)$ as a classifying space for $\mathrm{PVA}[Γ]$. To achieve this, we establish a connection between $Ω(Γ)$ and the Salvetti complex associated with a specific Coxeter graph $\widehatΓ$ related to $Γ$, showing that they share a common covering space. This finding links the asphericity of $Ω(Γ)$ to the $K(π, 1)$-conjecture for Artin groups associated with $\widehatΓ$. Additionally, the paper introduces and studies almost parabolic (AP) reflection subgroups, which play a crucial role in constructing these complexes.