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 $\hat{I}“$, we define a CW-complex $\hat{I}\text{c◯}(\hat{I}“)$ whose fundamental group is isomorphic to the pure virtual Artin group $\mathrm{PVA}[\hat{I}“]$, which coincides with the pure virtual braid group when $\hat{I}“$ 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 $\hat{I}\text{c◯}(\hat{I}“)$ and demonstrate that it holds when $\hat{I}“$ is of spherical type or of affine type, thereby characterizing $\hat{I}\text{c◯}(\hat{I}“)$ as a classifying space for $\mathrm{PVA}[\hat{I}“]$. To achieve this, we establish a connection between $\hat{I}\text{c◯}(\hat{I}“)$ and the Salvetti complex associated with a specific Coxeter graph $\hat{\hat{I}}“$ related to $\hat{I}“$, showing that they share a common covering space. This finding links the asphericity of $\hat{I}\text{c◯}(\hat{I}“)$ to the $K(\ddot{I}€,1)$-conjecture for Artin groups associated with $\hat{\hat{I}}“$. Additionally, the paper introduces and studies almost parabolic (AP) reflection subgroups, which play a crucial role in constructing these complexes.

71 pages, 17 figures