关于特征3中稳定等价于9维量子完全交的块和阿贝尔亏群猜想的一个例子

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…

1-上同调群的一些计算和非轨道等价作用量的构造

Some computations of 1-cohomology groups and construction of non orbit equivalent actions

对于每个群$G$都有一个具有相对性质（T）的无穷正规子群（例如$G = H \times K$，其中$H$是具有性质（T）的无穷正规子群，而$K$是任意的），以及任何可数阿贝尔群$Λ$，我们构造了$G$在概率空间上的自由遍历测度保持作用$σ_Λ$，使得$σ_Λ$的第一上同调群$H^1(σ_Λ)$等于Char $(G) \times Λ$ .我们推导出$G$有无数个非稳定轨道等价作用量。我们还计算了1-上同调群，并证明了群的自由积存在“多”非稳定轨道等价作用。

Regular subgroups with large intersection

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have recently raised for their applications in symmetric cryptography.

自由阿贝尔乘自由群的数学问题

Algorithmic problems for free-abelian times free groups

我们研究直积的自由交换和自由群，特别强调算法问题。在给出标准概念到该族的自然扩张后，我们找到了$\ZZ^m \times F_n$的任意自同态的显式表达式。这些工具用于解决$\ZZ^m \times F_n $的几个算法和判定问题：成员问题，同构问题，有限指标问题，子群和陪集相交问题，不动点问题，和Whitehead问题。

有限群的交换子群分类空间的同伦余极限

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

拓扑群$G$的分类空间BG可以通过使用自由群的降序中心列的子空间序列$B(q,G)$来过滤。如果$G$是有限的，在应用同伦理论方法时，将它们描述为同伦余极限是方便的。本文引入了$B(q,G)$的自然子空间$B(q,G)_p$，定义了固定素数$p$ .我们表明，$B(q,G)$是稳定的同伦等价于一个楔形的$B(q,G)_p$ $p$运行在素数划分的顺序$G$。阿贝尔群的上极限在理解这些空间的同伦类型中起着重要的作用。超特殊的$2$ -群是关键的例子，它们的共极限是有限的。证明了对于特殊2-群，$B(2,G)$不具有$K(π,1)$空间的同伦型.对于有限群$G$，我们计算了$B(2,G)$模挠的复K-理论。

The intersection graph of a finite simple group has diameter at most 5

Let $G$ be a non-abelian finite simple group. In addition, let $Δ_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of $Δ_G$ has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

On the lower central series of Baumslag-Solitar groups

We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has length 2. For some Baumslag-Solitar groups a connection is found between the intersection of all subgroups of finite index and the intersection of all terms of the lower central series.

直角Coxeter群的外自同构群要么是大的，要么是几乎交换的

Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

我们推广的概念，一个分离的交叉链接（SIL）给必要和充分的标准上的定义图$Γ$的直角Coxeter组$W_Γ$，使其外部自同构群是大的：也就是说，它包含一个有限的指数子群，承认自由群$F_2$作为商。当$Out(W_Γ)$不是很大时，我们证明了它实际上是阿贝尔的。我们还证明了有限阿贝尔群的图积的外自同构群也具有同样的二分法。因此，这些群具有性质（T）当且仅当它们是有限的，或者等价地$Γ$不包含SIL。

群的循环子群的交图

Intersection graph of cyclic subgroups of groups

设$G$是一个群。$G$的循环子群的交图（记为$\mathscr I_c(G)$）是以$G$的所有真循环子群为顶点的图，并且$\mathscr I_c(G)$中的两个不同顶点相邻当且仅当它们的交是非平凡的.本文对循环子群的交图为全不连通的、完全、星、路、圈之一的有限群进行了分类。我们证明了对给定的有限群$G$，$girth(\mathscr I_c (G)) \in \{3, \infty\}$ .此外，我们对所有循环子群的交图为平面的有限非循环交换群进行了分类。对于任意群$G$，我们确定了$\mathscr I_c (G)$的独立数、团覆盖数，并证明了$\mathscr I_c (G)$是弱$α$ -完美的.在其他结果中，我们确定了$\mathscr I_c (\mathbb{Z}_n)$是正则的$n$的值，并估计了它的控制数。

virtually abelian 群的扭曲共轭增长

The twisted conjugacy growth of virtually abelian groups

本文研究了 virtually abelian 群上与扭曲共轭相关的多个增长函数的渐近性，并证明这些增长函数均表现出多项式渐近行为。

Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether $H \cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H \cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian grou…

关于有限阿贝尔群的能力

On capability of finite abelian groups

Baer通过不变因子分解中因子大小的条件（群必须是非循环的，并且前两个不变因子必须相等）刻画了有限阿贝尔群（如果一个群同构于某个群的中心商，则该群是有能力的）。我们提供了一个不同的表征，在格的子群的条件。即有限阿贝尔群G是可交换群当且仅当存在G的子群族{H_i}具有平凡交，使得并生成G，且所有平凡交G/H_i具有相同的指数.子群族生成G的条件可以用子群族覆盖G的条件代替，而同幂元具有相同指数的条件可以用同幂元同构的条件代替。

On the stable Andreadakis problem

Let $F\_n$ be the free group on $n$ generators. Consider the group $IA\_n$ of automorpisms of $F\_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA\_n$: the first one is its lower central series $Γ\_*$; the second one is the Andreadakis filtration $\mathcal A\_*$, defined from the action on $F\_n$. In this paper, we establish that the canonical morphism between the associated graded Lie rings ${\mathcal L}(Γ\_*)$ and ${\mathcal L}(\mathcal A\_*)$ is stably surjective. We then investigate a $p$-restricted version of the Andreadakis problem. A calculation of t…