Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

The virtual braid group $V{B}_n$, the virtual twin group $V{T}_n$ and the virtual triplet group $V{L}_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group $S_n$ are the pure virtual braid group $V{P}_n$, the pure virtual twin group $PV{T}_n$ and the pure virtual triplet group $PV{L}_n$, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group $PV{L}_n$, the commutator subgroup $V{T}_n^{\prime }$ of $V{T}_n$ and the commutator subgroup $V{L}_n^{\prime }$ of $V{L}_n$. Our results complete the understanding of these groups, except that of $V{B}_n^{\prime }$, for which the existence of a finite presentation is not known for $n\ge 4$. We also prove that $V{L}_n/PV{L}_n^{\prime }$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.