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Spaces Related to Virtual Artin Groups

Federica Gavazzi

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.

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 , we define a CW-complex whose fundamental group is isomorphic to the pure virtual Artin group , which coincides with the pure virtual braid group when is . This construction generalizes the previously studied BEER complex, originally defined for pure virtual braids, to all Coxeter graphs. We investigate the asphericity of and demonstrate that it holds when is of spherical type or of affine type, thereby characterizing as a classifying space for . To achieve this, we establish a connection between and the Salvetti complex associated with a specific Coxeter graph related to , showing that they share a common covering space. This finding links the asphericity of to the -conjecture for Artin groups associated with . Additionally, the paper introduces and studies almost parabolic (AP) reflection subgroups, which play a crucial role in constructing these complexes.

Paper Structure

This paper contains 17 sections, 53 theorems, 97 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter groups and Artin groups
  4. Coxeter polytopes
  5. Virtual Artin groups
  6. The Salvetti complex and the AP reflection subgroups
  7. The Salvetti complex for TEXT
  8. Almost parabolic reflection subgroups
  9. The Salvetti complex of TEXT
  10. The BEER space TEXT
  11. The Complex TEXT and the failure of the CAT(0) Property
  12. The generalized BEER complex
  13. The common covering of TEXT and TEXT
  14. The group TEXT and the two regular coverings
  15. The covering space TEXT
  16. ...and 2 more sections

Key Result

Theorem \ref{FalseCat0}

The complex $\Omega_3$ is not locally CAT(0).

Figures (9)

  • Figure 1: The Coxeter polytope for $X={\{s,t\}}$ and $m_{s,t}=4$.
  • Figure 2: The 2-cell $\Delta^2(X)$ of the Salvetti complex $\overline{Sal}(\Gamma)$.
  • Figure 3: The two polytopes $C[\mathcal{X}]$ and $C[\mathcal{X}']$.
  • Figure 4: The Permutohedron for $n=3$.
  • Figure 5: The Permutohedron for $n=3$ with the associated ordered partitions.
  • ...and 4 more figures

Theorems & Definitions (110)

  • Theorem \ref{FalseCat0}
  • Corollary \ref{beeraspht}
  • Corollary \ref{originalbeeraspht}
  • Corollary \ref{beerasphtAFF}
  • Theorem \ref{maintheorem}
  • Theorem 2.2
  • Theorem 2.3: Cox34,Cox35
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 100 more