Virtual braid groups, virtual twin groups and crystallographic groups

Let $n\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $\Pi$ be one of the following quotients: $VB_n/\Gamma_2(VP_n)$ or $VT_n/\Gamma_2(PVT_n)$ where $\Gamma_2(H)$ is the commutator subgroup of $H$. In this paper, we show that $\Pi$ is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in $\Pi$. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in $\Pi$. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case.Comment: In this new version some general results were added in Section