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

Abstract

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 $V{B}_n$. In particular, we give a solution to the word problem in $V{B}_n$, and we prove that the virtual cohomological dimension of $V{B}_n$ is $n-1$.