Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

Pub Date : 2024-05-17 DOI:10.1016/j.jpaa.2024.107713

Pravin Kumar , Tushar Kanta Naik , Neha Nanda , Mahender Singh

引用次数: 0

Abstract

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 $P V T_{n}$ and the pure virtual triplet group $P V L_{n}$ , 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 $P V L_{n}$ , the commutator subgroup $V T_{n}^{'}$ of $V T_{n}$ and the commutator subgroup $V L_{n}^{'}$ of $V L_{n}$ . Our results complete the understanding of these groups, except that of $V B_{n}^{'}$ , for which the existence of a finite presentation is not known for $n \geq 4$ . We also prove that $V L_{n} / P V L_{n}^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.

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对称群虚扩展的换元子群和晶商

虚辫群 VBn、虚孪群 VTn 和虚三重群 VLn 是对称群 Sn 的扩展，它们是由虚结理论的亚历山大-马尔科夫对应关系激发的。这些群到对称群 Sn 的自然外变形的内核分别是纯虚辫群 VPn、纯虚孪生群 PVTn 和纯虚三重群 PVLn。本文研究了这些群的换元子群、纯子群和结晶商。我们推导了纯虚三重群 PVLn、VTn 的换元子群 VTn′ 和 VLn 的换元子群 VLn′ 的显式有限呈现。我们的结果完成了对这些群的理解，但 VBn′ 除外，因为对于 n≥4 的 VBn′，有限呈现的存在尚不可知。我们还证明了 VLn/PVLn′ 是一个结晶群，并给出了其中无限多个扭元的明确构造。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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