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Arithmetic properties of the cohomology of Artin groups. (English) Zbl 0973.20025

Let \(\Delta\) be a spherical Dynkin diagram and let \(Br(\Delta)\) be the braid group associated to this Dynkin diagram. In the paper under review the authors compute the cohomology group \(H^n(Br(\Delta),\mathbb{Q}[q,q^{-1}])\) for any \(n\). The action of \(Br(\Delta)\) on \(\mathbb{Q}[q,q^{-1}]\) is defined by multiplication by \(-q\) for each of the standard generators of the generalized braid group \(Br(\Delta)\).
In an earlier paper the first three authors already determined the case of \(\Delta\) being \(A_n\) which leads to the classical braid group. The proof there used an explicit combinatorial description of the classifying space of \(Br(A_n)\) which was given by Salvetti. The classifying space is deduced from the Tits building of the corresponding Coxeter group. The case \(\Delta=B_n\) and \(\Delta=D_n\) is treated by the fact that these Dynkin diagrams have subdiagrams of type \(A_k\) for smaller \(k\) and therefore the Tits building of \(B_n\) and \(D_n\) contains the building of type \(A_k\). The classifying spaces for these groups can be linked this way and using the combinatorial formula of Salvetti as well as the result for Dynkin diagrams of type \(A_{n-1}\) one gets a complete description of the additive structure of these cohomology groups.
The exceptional cases \(I_2(m)\), \(H_3\), \(H_4\), \(F_4\), \(E_6\), \(E_7\) and \(E_8\) are computed by a computer program. Details are not given.
Both, the proof for type \(B_n\) and \(D_n\) and the final result are quite complicated and involved.

MSC:

20F36 Braid groups; Artin groups
20J06 Cohomology of groups
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References:

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