论文标题
纯虚拟双胞胎组的结构和自动形态
Structure and automorphisms of pure virtual twin groups
论文作者
论文摘要
在没有三重或更高相交的封闭式表面上的有限浸泡圆圈集合的稳定同位素类别的研究被认为是虚拟结理论的平面类似物，这是经典结理论的远程概括。最近的作品在平面环境中建立了亚历山大和马尔可夫定理。在经典的情况下，小组的作用是由双胞胎组（一类右角Coxeter组）扮演的。一个新的称为虚拟双胞胎组的新组以自然的方式扩展了双胞胎组，在虚拟情况下扮演组的角色。虚拟双胞胎组$ vt_n $包含纯虚拟双胞胎$ pvt_n $，这是纯Artin Braid group的平面类似物。在本文中，我们证明了纯虚拟双胞胎组$ pvt_n $是一个不可约束的右角Artin Group，它具有琐碎的中心，并提供了精确的表现。我们表明，$ pvt_n $是无限排名组的迭代半独立产品的分解。我们对$ pvt_n $的自动形态组进行完整描述，并建立自然序列组的自然精确序列。作为应用程序，我们表明$ vt_n $是残留有限的，$ pvt_n $具有$ r_ \ infty $ -property。
Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group $VT_n$ contains the pure virtual twin group $PVT_n$, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group $PVT_n$ is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that $PVT_n$ has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of $PVT_n$ and establish splitting of natural exact sequences of automorphism groups. As applications, we show that $VT_n$ is residually finite and $PVT_n$ has the $R_\infty$-property.
