Virtual braid groups, virtual twin groups and crystallographic groups

Abstract

Let n≥2. Let VBn (resp. VPn) be the virtual braid group (resp. the pure virtual braid group), and let VTn (resp. PVTn) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients: VBn/Γ2(VPn) or VTn/Γ2(PVTn) where Γ2(H) is the commutator subgroup of H. In this paper, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) modulo the respective commutator subgroup in each case.