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Crystallographic groups and flat manifolds from complex reflection groups. (English) Zbl 1353.20020
The paper is a generalization of an article by D. L. Gonçsalves, J. Guaschi and O. Ocampo [“Quotients of the Artin braid groups and crystalographic groups”, Preprint, arxiv:1503.04527].
In that paper, the authors proved that the quotient of the braid group \(B_n\) by the commutator subgroup \([P_n, P_n]\) of the pure braid group is a crystallographic group. Morever, they proved that this quotient has no 2-torsion.
The author of the paper under review proves that all the results of the above article can be generalized for the case where the group \(B_n\) is a generalized braid group \(B\) of a (finite) complex reflection group in the sense of an article by M. Broúe et al. [J. Reine Angew. Math. 500, 127–190 (1998; Zbl 0921.20046)]. The author considers the quotient \(B/[P, P]\), where \(P\) is the pure braid group. He proves that this quotient is always a crystallographic group, and that it never contains elements of order 2. Another result is a construction of elements of finite order inside \(B/[P, P]\). In this part, torsion-free examples of Bieberbach groups in the above way are given and the question which one has a Kähler structure is considered.

20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20H15 Other geometric groups, including crystallographic groups
Full Text: DOI arXiv
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