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

Abstract

Let $Γ$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,Σ$) 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 $E_Γ$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a simplicial complex $Ω(Γ)$ having the same homotopy type as $E_Γ$. We observe that, if $T \subset S$, then $Ω(Γ_T)$ naturally embeds into $Ω(Γ)$. We prove that this embedding admits a retraction $π_T: Ω(Γ) \to Ω(Γ_T)$, 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 $Φ$, we show that $Φ$ has the same homotopy type as the universal cover of $E_Γ$, and we prove that $Φ$ is CAT(0) if and only if $\SS$ is a flag complex. We say that $X \subset S$ is free of infinity if $Γ_X$ has no edge labeled by $\infty$. We show that, if $E_{Γ_X}$ is aspherical and $A_X$ has a solution to the word problem for all $X \subset S$ free of infinity, then $E_Γ$ 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 $n-1$.