Crystallographic groups and flat manifolds from complex reflection groups (Q292067)

The paper is a generalization of an article by \textit{D. L. Gonçsalves}, \textit{J. Guaschi} and \textit{O. Ocampo} [``Quotients of the Artin braid groups and crystalographic groups'', Preprint, \url{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 \textit{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.