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 EΓ 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 EΓ. We observe that, if T ⊂ S, then Ω(ΓT) naturally embeds into Ω (Γ). We prove that this embedding admits a retraction πT: Ω(Γ) → Ω(ΓT), and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family 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 EΓ, and we prove that Φ is CAT(0) if and only if S is a flag complex. We say that X ⊂ S is free of infinity if ΓX has no edge labeled by ∞. We show that, if is aspherical and AX has a solution to the word problem for all X ⊂ S free of infinity, then EΓ is aspherical and A has a solution to the word problem. We apply these results to the virtual braid group VBn. In particular, we give a solution to the word problem in VBn, and we prove that the virtual cohomological dimension of VBn is n-1. © 2012 Springer-Verlag.