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Let $\Gamma$ be a Coxeter graph, let $(W,S)$ be its associated Coxetersystem, 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 Titscone, and let $E_\Gamma$ be the complement in $I +iV$ of the reflectinghyperplanes. Recall that Charney, Davis, and Salvetti have constructed asimplicial 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 aretraction $\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)$, and we deduce severaltopological and combinatorial results on parabolic subgroups of $A$. From afamily $\SS$ of subsets of $S$ having certain properties, we construct a cubecomplex $\Phi$, we show that $\Phi$ has the same homotopy type as the universalcover of $E_\Gamma$, and we prove that $\Phi$ is CAT(0) if and only if $\SS$ isa flag complex. We say that $X \subset S$ is free of infinity if $\Gamma_X$ hasno 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 ofinfinity, then $E_\Gamma$ is aspherical and $A$ has a solution to the wordproblem. We apply these results to the virtual braid group $VB_n$. Inparticular, we give a solution to the word problem in $VB_n$, and we prove thatthe virtual cohomological dimension of $VB_n$ is $n-1$.

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