Z. Araghi Rostami, M. Parvizi, P. Niroomand
In this paper we extend the notion of CP covers for groups to the field of Lie algebras, and show that despite the case of groups, all CP covers of a Lie algebra are isomorphic. Finally we show that CP covers of groups and Lie rings which are in Lazard correspondence, are in Lazard correspondence too, and the Bogomolov multipliers are isomorphic.
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Computing Bogomolov multipliers and finitary Bogomolov multipliers of infinite groups
本文研究无限群的Bogomolov乘子，重点是多循环群，给出类二幂零群等结构描述，并提出两种在GAP中实现的算法。
In this paper, the Bogomolov multiplier of $p$-groups of order $p^7$ ($p>2$) and exponent $p$ is given.
The Bogomolov multiplier of a multiplicative Lie algebra
本文为乘法李代数引入Bogomolov乘子概念，建立Hopf型公式，并证明同斜乘法李代数的Bogomolov乘子同构。
We prove that if $G$ is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of $G$ in the following four cases: (i) $G$ is metabelian, (ii) $\exp G=4$, (iii) $G$ is nilpotent of class $\le 5$, or (iv) $G$ is a $4$-Engel group.
We introduce the $q$-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under $q$-isoclinism. We prove that the $q$-Schur Multiplier is invariant under $q$- exterior isoclinism, and as an easy consequence we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if $G$, $H$ are $p$-groups and $G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$, then the cardinality of the minimal number of generators of $G$ and $H$ are the same. Moreover we prove some structural results about $q$-nonabelian tensor square of groups.
The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology classes which vanish after restricting to every abelian subgroup of $G$. We give a new proof of a Hopf-type formula for $B_0(G)$ and derive an exact sequence for the cohomological version of the Bogomolov multiplier. Using this exact sequence we provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map. Finally, we give a complete list of groups of order $p^6$, for odd…
Universal commutator relations, Bogomolov multipliers, and commuting probability
设$G$是有限$p$ -群.证明了当$G$的交换概率大于$(2p^2 + p - 2)/p^5$时，$G$ -不变函数域的非分歧Brauer群是平凡的.等价地，$G$中的所有映射之间的关系都是某些普适关系的结果。该界是最佳可能的，并给出了所有有限群的$1/4$的全局下界.这个结果是通过描述Bogomolov乘子非平凡且其所有真子群和真子群的Bogomolov乘子平凡的群的结构得到的.应用包括分类的$p$ -组的最低阶有非平凡的Bogomolov乘数和幂零类$2$，一个非概率标准消失的Bogomolov乘数，并建立一个序列的Bogomolov的绝对$γ$ -最小的因素，这是$2$ -组的任意大幂零类，从而提供反例的一些Bogomolov的索赔。在这方面，我们填补了空白的证明平凡的有限单群的Bogomolov乘子。
Local control and Bogomolov multipliers of finite groups
本文研究有限群 Bogomolov 乘子的局部控制问题，证明在 Sylow $p$-子群幂零类不超过 $p$ 时，其乘子的 $p$-部分具有局部受控性。
Bogomolov multipliers of all groups of order 128
本文是\cite{Mor11}与\cite{Eic08}结合给出的Bogomolov乘子算法的一个实现，以有效地确定$128$阶群的乘子。两个服务目的是结果\cite{Chu08,Chu09}的延续和应用程序\cite{Jez13}。
Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier $B_0(G)$ is trivial in terms of the lower central series of $G$ and $P_1 = C_G(γ_2(G) / γ_4(G))$. If in addition $G$ has positive degree of commutativity and $P_1$ is metabelian, we show how understanding $B_0(G)$ reduces to the simpler commutator structure of $P_1$. This result covers all $p$-groups of maximal class of large enough order and, furthermore, it allows us to give the first natural family of $p$-groups containing an abundance of groups with nontrivial Bogomolov multipliers…
On some closure properties of the non-abelian tensor product and the Bogomolov multiplier
证明了有限幂零类、有限可解类、有限多环类、幂零类$n$的幂零类和超可解群在非交换张量积的构造下是封闭的.给出了有限生成群的非交换张量积为有限生成群的充要条件。证明了大多数有限单群的中心扩张都有平凡的Bogomolov乘子.
Units of group rings, the Bogomolov multiplier, and the fake degree conjecture
设$π$是有限$p$ -群，$\mathbb{F}_q$是有$q=p^n$元的有限域.用$\mathrm{I}_{\mathbb{F}_q}$表示群环$\mathbb{F}_q[π]$的增广理想。我们发现了$1+\mathrm{I}_{\mathbb{F}_q}$的阿贝尔化、$π$的Bogomolov乘子$\mathrm{B}_0(π)$和$π$的共轭类数$\mathrm{k}(π)$之间的惊人关系：\[ | (1+\mathrm{I}_{\mathbb{F}_q})_{\mathrm{ab}} |=q^{\mathrm{k}(π)-1}|\mathrm{B}_0(π)|. \]特别地，如果$π$是具有非平凡Bogomolov乘子的有限$p$ -群，则$1+\mathrm{I}_{\mathbb{F}_q}$是M.艾萨克
Bogomolov multipliers of word labelled oriented graph groups
本研究针对字标记有向图（WLOG）群，探讨其同调版本 Bogomolov 乘子 $\tilde{B_0}$，证明可通过底层 WLOG 计算生成元，将 Bestvina-Brady 群和 Artin 群归为 WLOG 群并计算两种乘子，为前者 Schur 乘子结构提供新证明。
The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3,4).
Bogomolov multipliers for some $p$-groups of nilpotency class 2
有限群$G$的Bogomolov乘子$B_0(G)$被定义为Schur乘子的子群，该子群由限制到$G$的所有交换子群后消失的上同调类组成。Bogomolov乘数的平凡性是Noether问题的障碍。证明了若$G$是$G_1$和$K_i\leq Z(G_i), i=1,2$的中心积，且$θ:G_1\to G_2$是一个群同态，使得它的限制$θ\vert_{K_1}:K_1\to K_2$是一个同构，则$B_0(G_1/K_1), B_0(G_1)$和$B_0(G_2)$的平凡性蕴涵了$B_0(G)$的平凡性.我们给出了一个积极的答案Noether的问题，所有$2$ -发电机$p$ -群的幂零类$2$，并为一系列的$4$ -发电机$p$ -群的幂零类$2$（通常的要求为根的单位）。
The Bogomolov multiplier of rigid finite groups
有限群$G$的Bogomolov乘子被定义为Schur乘子的子群，该Schur乘子由限制到$G$的所有交换子群后消失的上同调类组成.这个不变量在商空间$G$的双有理几何中起着重要的作用。我们表明，在许多情况下，消失的Bogomolov乘数是保证刚性的$G$在这个意义上，它没有外部类保持自同构。
$\tilde{B_0}$-invariant of groups
该研究针对群簇推广了1988年提出、2012年被给出等价定义的Bogomolov乘子，得到了相关主题的若干新结果。
Unramified Brauer groups of finite and infinite groups
Bogomolov乘子是同构于给定商空间的非分歧Brauer群的群理论不变量。我们推导出一个同调版本的Bogomolov乘数，证明了一个霍普夫型公式，找到一个五项精确序列对应于这个不变量，并描述了中心扩展理论中的Bogomolov乘数的作用。给出了第二类幂零群的Bogomolov乘子的一个新刻画.我们定义的Bogomolov乘数在K-理论和证明其平凡性是等价于解决一个长期存在的问题所造成的低音。提出了一种计算Bogomolov乘子的算法。
Bogomolov multipliers for unitriangular groups
有限群$G$的Bogomolov乘子$B_0(G)$被定义为Schur乘子的子群，该子群由限制到$G$的所有交换子群后消失的上同调类组成。本文对Kang和Kunyavskiovskiy在\cite{KK}中提出的一个公开问题给出了肯定的回答.也就是说，我们证明了：如果$G$是$\f$上的单三角群，或者是它的下中心列的商，或者是它的下中心列的子群，或者是两个单三角群的中心积，则$B_0(G)=0$ .
Let $G$ be a finite group, $V$ a faithful finite-dimensional representation of $G$ over the complex field $\mathbb{C}$ and $\mathbb{C}(V)^{G}$ be the corresponding invariant field. The Bogomolov multiplier $B_{0}(G)$ of $G$ is canonically isomorphic to the unramified cohomological group $H_{\textrm{nr}}^{2}(\mathbb{C}(V)^{G},\mathbb{Q}/\mathbb{Z})$, which has been used by Saltman (1984) and Bogomolov (1988) to provide counter-examples to the rationality problem of $\mathbb{C}(V)^{P}$ for finite $p$-groups $P$ over $\mathbb{C}$. In this paper, we investigate the vanishing property of $B_{0}(P)$…
Topological perspectives on the vanishing of some Bogomolov multipliers
本文从拓扑视角出发，结合组合拓扑技术与Ihara-Yokonuma计算，证明有限单群和有限Coxeter群的博戈莫洛夫乘子均消失，为搭建其在代数几何与配边群中作用的桥梁提供了小步进展。
The Bogomolov multiplier of a group is the unramified Brauer group associated to the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated to nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups…
Commutativity preserving extensions of groups
在平行的经典理论的中心扩展的群体，我们开发了一个版本的扩展，保持交换性。结果表明，Bogomolov乘子是一个泛对象parametrising这样的扩展的一个给定的组。最大和最小的扩展检查，并探索与通勤概率的连接。这些考虑产生了博戈莫洛夫乘数的指数和秩的界限。
Unramified Brauer groups and isoclinism
证明了如果$G_1$和$G_2$是等倾群，则它们的Bogomolov乘子同构.
Representation Growth
本文的主要结果是关于一类群的表示增长的。在第$1$章中，我们提出了所需的初步理论。在第四章$2$中，我们介绍了定义在整体域$k$上的代数群$G$的同余子群问题。在$3$一章中，我们考虑$Γ=G(\mathcal{O}_S)$是半单代数$k$ -群的一个算术子群，它是一个整体域$k$的一个含$S$ -整数环$\mathcal{O}_S$。如果$G$的李代数是完备的，Lubotzky和Martin证明了如果$Γ$具有弱同余子群性质，则$Γ$具有多项式表示增长，即对于某个多项式$p$，有$r_n(Γ)\leq p(n)$。通过使用不同的方法，我们表明，同样适用于任何半单代数群$G$，包括那些与非完美李代数。在$4$一章中我们证明了如果$Γ$具有弱同余子群性质，则对于某个常数$D$，有$s_n(Γ)\leq n^{D\log n}$，其中$s_n(Γ)$表示$Γ$的指数至多为$n$的子群的个数.在$5$一章中我们考虑$Γ=1+J$，其中$J$是一个有限幂零结合代数，这被称为一个代数群。我们通过观察$Γ=1+I_{\mathbb{F}_q}$形式的群，为假度猜想提供了任何素数$p$的反例，其中$I_{\mathbb{F}_q}$是某些$p$ -群$π$的群代数$\mathbb{F}_q[π]$的增广理想。此外，我们表明，对于这样的群体$r_1(Γ)=q^{K(π)-1}|B_0(π)|$，其…
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