Oriented pro-$\ell$ groups with the Bogomolov-Positselski property

On some closure properties of the non-abelian tensor product and the Bogomolov multiplier

We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.

Units of group rings, the Bogomolov multiplier, and the fake degree conjecture

Let $π$ be a finite $p$-group and $\mathbb{F}_q$ a finite field with $q=p^n$ elements. Denote by $\mathrm{I}_{\mathbb{F}_q}$ the augmentation ideal of the group ring $\mathbb{F}_q[π]$. We have found a surprising relation between the abelianization of $1+\mathrm{I}_{\mathbb{F}_q}$, the Bogomolov multiplier $\mathrm{B}_0(π)$ of $π$ and the number of conjugacy classes $\mathrm{k}(π)$ of $π$: \[ | (1+\mathrm{I}_{\mathbb{F}_q})_{\mathrm{ab}} |=q^{\mathrm{k}(π)-1}|\mathrm{B}_0(π)|. \] In particular, if $π$ is a finite $p$-group with a non-trivial Bogomolov multiplier, then $1+\mathrm{I}_{\mathbb{F}_…

The Bogomolov multiplier of finite simple groups

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).

有限群与无限群的非分歧Brauer群

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。

字标记有向图群的 Bogomolov 乘子

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。

Bogomolov multipliers for some $p$-groups of nilpotency class 2

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. We show that if $G$ is a central product of $G_1$ and $G_2$, regarding $K_i\leq Z(G_i), i=1,2$, and $θ:G_1\to G_2$ is a group homomorphism such that its restriction $θ\vert_{K_1}:K_1\to K_2$ is an isomorphism, then the triviality of $B_0(G_1/K_1), B_0(G_1)$ and $B_0(G_2)$ implies the triviality of $B_0(G)$. We …

Bogomolov multipliers of some groups of order $p^{6}$

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 $B_{0}(P)$…

单三角群的Bogomolov乘子

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$B_0(G)=0$ .

Mild pro-p groups and the Koszulity conjectures

Let $p$ be a prime, and $\mathbb{F}_p$ the field with $p$ elements. We prove that if $G$ is a mild pro-$p$ group with quadratic $\mathbb{F}_p$-cohomology algebra $H^\bullet(G,\mathbb{F}_p)$, then the algebras $H^\bullet(G,\mathbb{F}_p)$ and $\mathrm{gr}\mathbb{F}_p[\![G]\!]$ - the latter being induced by the quotients of consecutive terms of the $p$-Zassenhaus filtration of $G$ - are both Koszul, and they are quadratically dual to each other. Consequently, if the maximal pro-$p$ Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.

Anabel相交理论I：Bogomolov-Pop猜想及其应用

Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications

完成了F猜想的证明。Bogomolov和F. Pop：设$F_{1}$和$F_{2}$是有限生成的域，分别在$k_{1}$和$k_{2}$上具有超越度$\geq 2$，其中$k_{1}$是$\bar{\mathbb{Q}}$或$\bar{\mathbb{F}}_{p}$，$k_{2}$是代数闭的。我们用$G_{F_1}$和$G_{F_2}$表示它们各自的绝对伽罗瓦群。则从$F_1$和$F_2$的不可分闭包的同构直到Frobenius扭到它们的伽罗瓦群的连续外同构的正则映射$φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$是双射。因此，在素域的代数闭包上的维数为$\geq 2$的簇的函数域是Anabel的。我们应用这一点给出了Grothendieck-Teichmüller群的元素成为$\bar{\mathbb{Q}}$的绝对Galois群的元素的一个充分必要条件。

One relator maximal pro-p Galois groups and the Koszulity conjectures

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p)<\infty$ and $\dim H^2(G_{K},\mathbb{F}_p)=1$, we show that L.~Positselski's and T.~Weigel's Koszulity conjectures are true for ${K}$. Also, under the above hypothesis we show that the $\mathbb{F}_p$-cohomology algebra of $G_{K}$ is the quadratic dual of the graded algebra $\mathrm{gr}_\bullet\mathbb{F}_p[G_{K}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{K}]$, and these two algebras decom…

Irrationality of generic quotient varieties via Bogomolov multipliers

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

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrising such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.