Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916280426037248 |
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| author | Kumar, Pravin Naik, Tushar Kanta Nanda, Neha Singh, Mahender |
| author_facet | Kumar, Pravin Naik, Tushar Kanta Nanda, Neha Singh, Mahender |
| contents | The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ 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 $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, 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 $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09804 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups Kumar, Pravin Naik, Tushar Kanta Nanda, Neha Singh, Mahender Group Theory Geometric Topology Primary 20F55, 20H15, Secondary 20F36 The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ 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 $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, 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 $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it. |
| title | Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups |
| topic | Group Theory Geometric Topology Primary 20F55, 20H15, Secondary 20F36 |
| url | https://arxiv.org/abs/2303.09804 |