20 research outputs found

    Cohomology of Artin groups of type tilde{A}_n, B_n and applications

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    We consider two natural embeddings between Artin groups: the group G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of type A_n. The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of G_{B_n} with coefficients over the module Q[q^{+-1},t^{+-1}], where the action is (-q)-multiplication for the standard generators associated to the first n-1 nodes of the Dynkin diagram, while is (-t)-multiplication for the generator associated to the last node. As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional representation obtained by Tong, Yang and Ma. We stress the topological significance, recalling some constructions of explicit finite CW-complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case. For affine groups, some of these orbit spaces are known to be K(pi,1) spaces (in particular, for type tilde{A}_n). We point out that the above cohomology of G_{B_n} gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

    T-structures on some local Calabi-Yau varieties

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    Let ZZ be a Fano variety satisfying the condition that the rank of the Grothendieck group of ZZ is one more than the dimension of ZZ. Let ωZ\omega_Z denote the total space of the canonical line bundle of ZZ, considered as a non-compact Calabi-Yau variety. We use the theory of exceptional collections to describe t-structures on the derived category of coherent sheaves on ωZ\omega_Z. The combinatorics of these t-structures is determined by a natural action of an affine braid group, closely related to the well-known action of the Artin braid group on the set of exceptional collections on ZZ.Comment: 30 page

    Braid groups of imprimitive complex reflection groups

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    We obtain new presentations for the imprimitive complex reflection groups of type (de,e,r)(de,e,r) and their braid groups B(de,e,r)B(de,e,r) for d,r≥2d,r \ge 2. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group B(de,e,r)B(de,e,r) is a semidirect product of the braid group of affine type A~r−1\widetilde A_{r-1} and an infinite cyclic group. Elements of B(de,e,r)B(de,e,r) are visualized as geometric braids on r+1r+1 strings whose first string is pure and whose winding number is a multiple of ee. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group B(de,e,r)B(de,e,r) is strongly translation discrete.Comment: published versio

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

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    Let Γ\Gamma be a Coxeter graph, let (W,S)(W,S) be its associated Coxeter system, and let (A,Σ(A,\Sigma) be its associated Artin-Tits system. We regard WW as a reflection group acting on a real vector space VV. Let II be the Tits cone, and let EΓE_\Gamma be the complement in I+iVI +iV of the reflecting hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a simplicial complex Ω(Γ)\Omega(\Gamma) having the same homotopy type as EΓE_\Gamma. We observe that, if T⊂ST \subset S, then Ω(ΓT)\Omega(\Gamma_T) naturally embeds into Ω(Γ)\Omega (\Gamma). We prove that this embedding admits a retraction πT:Ω(Γ)→Ω(ΓT)\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T), and we deduce several topological and combinatorial results on parabolic subgroups of AA. From a family \SS of subsets of SS having certain properties, we construct a cube complex Φ\Phi, we show that Φ\Phi has the same homotopy type as the universal cover of EΓE_\Gamma, and we prove that Φ\Phi is CAT(0) if and only if \SS is a flag complex. We say that X⊂SX \subset S is free of infinity if ΓX\Gamma_X has no edge labeled by ∞\infty. We show that, if EΓXE_{\Gamma_X} is aspherical and AXA_X has a solution to the word problem for all X⊂SX \subset S free of infinity, then EΓE_\Gamma is aspherical and AA has a solution to the word problem. We apply these results to the virtual braid group VBnVB_n. In particular, we give a solution to the word problem in VBnVB_n, and we prove that the virtual cohomological dimension of VBnVB_n is n−1n-1
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