Spaces Related to Virtual Artin Groups

TL;DR

This paper constructs a CW-complex for virtual Artin groups, generalizing previous complexes, and proves asphericity for certain types, linking it to the $K(\pi,1)$-conjecture for related Artin groups.

Contribution

It introduces a new CW-complex for virtual Artin groups, generalizes the BEER complex, and establishes asphericity results connecting to the $K(\pi,1)$-conjecture.

Findings

The complex $oldsymbol{oldsymbol{ ext{Ω}}(oldsymbol{ ext{ extGamma}})}$ is a classifying space for pure virtual Artin groups of spherical or affine type.

$oldsymbol{ ext{Ω}}(oldsymbol{ extGamma})$ shares a common covering space with the Salvetti complex of a related Coxeter graph.

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