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 Salvetti, Charney and Davis 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 $${\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)}$$ , and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family $${\mathcal{S}}$$ 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 $${\mathcal{S}}$$ is a flag complex. We say that $${X \subset S}$$ is free of infinity if Γ X has no edge labeled by ∞. 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 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.