$K(\pi,1)$ and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2010

$K(\pi,1)$ and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups

Eddy Godelle
  • Fonction : Auteur
  • PersonId : 873079
Luis Paris
  • Fonction : Auteur
  • PersonId : 862829

Résumé

Let $\Gamma$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,\Sigma$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and let $E_\Gamma$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a simplicial complex $\Omega(\Gamma)$ having the same homotopy type as $E_\Gamma$. We observe that, if $T \subset S$, then $\Omega(\Gamma_T)$ naturally embeds into $\Omega (\Gamma)$. We prove that this embedding admits a retraction $\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)$, and we deduce several topological and combinatorial results on parabolic subgroups of $A$. From a family $\SS$ of subsets of $S$ having certain properties, we construct a cube complex $\Phi$, we show that $\Phi$ has the same homotopy type as the universal cover of $E_\Gamma$, and we prove that $\Phi$ is CAT(0) if and only if $\SS$ is a flag complex. We say that $X \subset S$ is free of infinity if $\Gamma_X$ has no edge labeled by $\infty$. We show that, if $E_{\Gamma_X}$ is aspherical and $A_X$ has a solution to the word problem for all $X \subset S$ free of infinity, then $E_\Gamma$ is aspherical and $A$ has a solution to the word problem. We apply these results to the virtual braid group $VB_n$. In particular, we give a solution to the word problem in $VB_n$, and we prove that the virtual cohomological dimension of $VB_n$ is $n-1$.
Fichier principal
Vignette du fichier
TopCombArtV3.pdf (301.92 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00498766 , version 1 (08-07-2010)

Identifiants

Citer

Eddy Godelle, Luis Paris. $K(\pi,1)$ and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups. 2010. ⟨hal-00498766⟩
112 Consultations
45 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More