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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.

MSC:
 20F36 Braid groups; Artin groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20H15 Other geometric groups, including crystallographic groups
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References:
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