Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander–Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids and permutations, Ann. Inst. Fourier (Grenoble) 70 (2020), 3, 1341–1362], we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group VT n \mathrm{VT}_{n} on n ≥ 2 n\geq 2 strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group KT n \mathrm{KT}_{n} inside VT n \mathrm{VT}_{n} . As a by-product, it also follows that the twin group T n \mathrm{T}_{n} embeds inside the virtual twin group VT n \mathrm{VT}_{n} , which is an analogue of a similar result for braid groups.

中文翻译：

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虚拟平面编织组和排列

孪生组和虚拟孪生组分别是编织组和虚拟编织组的平面类似物。这些群在可定向表面上浸入圆的稳定同位素类理论的亚历山大-马尔可夫对应中扮演了辫子群的角色。受到 Artin 的总体想法以及 Bellingeri 和 Paris 最近的工作的启发 [P. Bellingeri 和 L. Paris，虚拟辫子和排列，安. 研究所。傅里叶（格勒诺布尔）70 (2020), 3, 1341–1362]，我们获得了虚拟孪生群和对称群之间同态的完整描述，作为一个应用，它为我们提供了虚拟孪生群自同构群的精确结构 室速 n \mathrm{VT}_{n} 在 n ≥ 2 n\geq 2 股。这是通过证明不可约直角 Coxeter 群的存在性来实现的 康泰 n \mathrm{KT}_{n} 里面 室速 n \mathrm{VT}_{n} 。作为副产品，双胞胎群体也随之而来 时间 n \mathrm{T}_{n} 嵌入虚拟双胞胎组内 室速 n \mathrm{VT}_{n} ，这是辫子组类似结果的模拟。