Conjugacy separability of some one-relator groups
D. Tieudjo, D. I. Moldavanskii
证明了由表示$<a, b; [a^m,b^n]=1>$（$m,n>1$）给出的单关系群类的任何群的共轭可分性.
Conjugacy separability of any group of the class of one-relator groups given by the presentation $<a, b; [a^m,b^n]=1>$ ($m,n>1$) is proven.
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One-relator groups with torsion are conjugacy separable
证明了具有挠率的单关系子群是遗传共轭可分的。我们的论点是基于Dani Wise和第一作者的最新结果的结合。作为推论，我们得到了具有挠率的单关系群的任何拟凸子群也是共轭可分的。
The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups
一般的{ \bf曲面群猜想}问一个单关系群是否是一个曲面群，其中每个有限指标的子群都是单关系群，并且每个无限指标的子群都是自由的（性质IF）。我们解决了[FKMRR]中给出的几个相关问题。首先，我们得到了循环压缩和共轭压缩单关系子群的曲面群猜想B。即：如果$G$是满足性质IF循环收缩单关系子群或共轭收缩单关系子群，则$G$是自由的、曲面群或可解的Baumslag-Solitar群。进一步将[FKMRR]中关于性质IF的结果与H. Wilton [W]和Stallings [St]，Kharlampovich和Myasnikov [KhM 4]的结果证明了[FKMRR]中提出的曲面群猜想C是成立的，即：如果$G$是一个具有IF性质的非自由生成的自由不可分解的完全剩余自由群，则$G$是一个曲面群.
Independence of the conjugacy problem and conjugacy separability
本文构造出共轭可分性函数任意大但共轭问题能多项式时间求解的有限生成群类，证明麦肯锡共轭问题算法复杂度可与最优解差任意远，揭示共轭问题与共轭可分性的独立性。
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy…
On generalized (twisted) conjugacy separability of some extensions of groups
引入广义共轭可分性性质，证明群$G$在有限与循环扩张下扭曲共轭、Brinkmann共轭等可分性的等价与蕴含关系，并应用于自由群等群类。
On the Conjugacy Separability of Generalized Free Products of Groups
证明了两个有限p-群的广义自由积是共轭p-可分群当且仅当它是剩余有限p-群。应用这个结果建立了无限群的广义自由积共轭p-可分性的若干充分条件。
There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.
We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are …
Subgroup Conjugacy Separability in Residually Free Groups
本文研究剩余自由群的子群共轭性质，证明有限展示情形具有子群共轭可分性，$FP_\infty$ 型还具可辨识性，并推出其外自同构群剩余有限。
On subgroup conjugacy separability in the class of virtually free groups
称群G为子群共轭可分群，如果G的任意两个非共轭生成的非共轭子群在G的某个有限商中保持非共轭。证明了有限群的有限树的自由群和基本群在某种正规化子条件下是SCS。引入子群的入共轭可分性，证明了上述群也具有这一性质。
Subgroup conjugacy separability for surface groups
一个群$G$称为子群共轭可分群（SCS），如果$G$的任意两个非共轭子群在$G$的某个有限商中保持非共轭.证明了可定向闭紧曲面的自由群和基本群是SCS。
Arbitrary residual finiteness and conjugacy separability growth
本文证明了Bradford构造的群具有共轭可分性且其增长等于剩余有限性增长，并扩展构造出新群，证明这两种增长函数能独立变化且可任意远离。
On conjugacy separability of graph products of groups
证明了当$\mathcal{C}$是有限群的扩张闭簇时，$\mathcal{C}$-遗传共轭可分群类在取任意图积下是闭的.作为结果，我们证明了$\mathcal{C}$-共轭可分群类在任意图积下是闭的。特别地，我们证明了直角Coxeter群是遗传共轭可分的和2-遗传共轭可分的，并且我们证明了无限生成的直角Artin群是遗传共轭可分的和对每个素数$p$都是$p$-遗传共轭可分的.
Hereditary conjugacy separability of right angled Artin groups and its applications
证明了直角Artin群的有限指数子群是共轭可分的.然后，我们应用这个结果来建立其他类别的群的各种性质。特别地，我们证明了任何字双曲Coxeter群包含一个共轭可分子群的有限指数，并有一个剩余有限的外自同构群。主要结果的另一个推论是Bestvina-Brady群是共轭可分的，并且存在可解的共轭问题。
On the conjugacy separability in the class of finite $p$-groups of finitely generated nilpotent groups
证明了对任意素数$p$，一个生成的幂零群在有限$p$ -群类中共轭可分当且仅当它的扭子群是有限$p$ -群且扭子群的商群是阿贝尔群.
Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion
我们得到了Coxeter群、Artin群和带挠的单关系群的自由性、拟凸性和可分性的一些结果。
Twisted conjugacy and separability
研究扭曲共轭分离性及其在群结构中的性质，证明多项式群由nilpotent群由有限群的完全扭曲共轭分离等价于所有商群余有限。
On the residual finiteness and other properties of (relative) one-relator groups
一个相对的单关系表示形式为P = < X，H ; R >，其中X是一个集合，H是一个群，R是X和H上的一个群词。我们证明了，如果从R中删除H中的所有项而得到的X上的群字具有我们所说的唯一极大极小性质，则由P定义的群是剩余有限的当且仅当H是剩余有限的（定理1）。我们应用这一点，以获得新的结果有关的剩余有限性（普通）单关系群（定理4）。我们还得到了关于单关系子群的共轭问题的结果（定理5），以及关于形式P的表示的相对非球面性的结果（定理6）。
Conjugacy p-separability of right-angled Artin groups and applications
证明了直角Artin群中p-幂指数的每个次正规子群是共轭p-可分的。作为应用，我们证明了任意直角Artin群在挠自由幂零群幂零群类中是共轭可分的.作为另一个应用，我们证明了直角Artin群的外自同构群是虚剩余p-有限的。我们还证明了直角群的Torelli群是剩余无挠幂零的，因此是剩余p-有限的和双序的。
On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms
证明了即使Coxeter图不含（4，4，2）三角形的Coxeter群也是共轭可分的.特别地，这适用于所有直角Coxeter群或字双曲甚至Coxeter群。对于任意Coxeter群W，我们还研究了产生相同抛物子群集合的Coxeter生成集之间的关系。作为应用，我们证明了如果W的自同构保持每个足够短的元素的共轭类，那么它是内部的。然后，我们得出的后果外自同构群的Coxeter群。
In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in…
Uniform Upper Bounds for Conjugacy Separability Growth in Nilpotent Groups
针对幂零群的共轭可分性增长函数，本文给出次数至多线性于幂零类、二次于Hirsch长度的显式多项式上界，并证明二步幂零情形下该界最优，同时纠正了文献错误。
A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. We prove that limit groups are subgroup conjugacy separable. We also prove this property for one relator groups of the form $R=\langle a_1,...,a_n\mid W^n\rangle$ with $n>|W|$. The property is also proved for virtual retracts (equivalently for quasiconvex subgroups) of hyperbolic virtually special groups.
A group $G$ is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if $G$ is a hyperbolic group, $H_1$ is a quasiconvex subgroup of $G$, and $H_2$ is a subgroup of $G$ which is elementwise conjugate into $H_1$, then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellat…
Virtual retractions, conjugacy separability and omnipotence
我们用圈积来证明群是共轭可分群或幂零群。这些标准是在循环子群的虚收缩。我们给出两个应用：一个简单的拓扑证明的定理Stebe的无限阶元素的富克斯群（第一类）是共轭区别，并证明表面群是万能的。
Effective conjugacy separability of virtually abelian groups
本文刻画了每个几乎阿贝尔群的共轭可分性函数，发现当扩张对应不可约表示时与剩余有限性函数相同，否则可能严格更大，并给出示例。
Conjugacy separability of 1-acylindrical graphs of free groups
本文利用群作用在有限树上的理论证明了有限的、由自由群组成的、具有共轭生成的棱道群的1-非圆柱图的基本群是共轭可分的。这有几个应用：我们证明，积极的，$C'(1/6)$单关系群是共轭可分的，我们提供了一个共轭可分版本的RIP建设，我们使用后者提供一个例子，两个提出的，剩余有限群有同构profinite完成，这样一个是共轭可分的，另一个甚至没有可解的共轭问题。
Conjugacy Distinguished Subgroups
设${\cal C}$是在取子群、同态像和扩张下闭的有限群的非空类.抽象剩余${\cal C}$群$R$的子群$H$称为共轭${\cal C}$ -区别的，如果无论何时$y\in R$，则$y$在$H$中有共轭当且仅当同样适用于$R$的每个商群$R/N\in {\cal C}$中的$y$和$H$的像。证明了在具有正规自由子群$Φ$且$R/Φ$在${\cal C}$中的群中，每个正规生成子群都是共轭${\cal C}$ -可区别的.我们还证明了极限群、Lyndon群和某些单关系子群（这里的${\cal C}$是所有有限群的类）的生成子群是共轭可区别的.
We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root $s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly stronger conditions that the word problem of $G$ is decidable. We also give conditions under which the question of which Magnus subgroups of $\…
On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups
该研究聚焦普通与广义Baumslag-Solitar（GBS）群的共轭可分性问题，在特定周期群类的封闭性条件下，推导非可解与可解GBS群的共轭可分判别准则，并得出GBS群共轭有限可分等价于剩余有限的推论。
Separability of double cosets and conjugacy classes in 3-manifold groups
设M = H^3 / Γ是有限体积的双曲三维流形。证明了若H和K是Γ的交换子群且g在Γ中，则双陪集HgK在Γ中可分.作为结果，我们证明了：如果M是一个闭的、可定向的Haken 3-流形，且M的环面分解的每一个双曲段的基本群是共轭可分的，那么M的基本群也是共轭可分的.推广了Agol和Wise最近的工作，得到了若M是紧致的可定向3-流形，则π_1（M）是共轭可分的.
Cohomological Separability of Baumslag--Solitar groups and Their Generalisations
本文在广义 Baumslag–Solitar 群类中给出上同调可分性的等价刻画，并对 Baumslag–Solitar 群得到上同调可分性与有限完备化上同调维数的三分法，从而构造出具有和不具有可分上同调的非剩余有限单关系群例子。
In this article, we completely characterize the asymptotic behavior of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.
We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are need to separate pairs of distinct conjugacy classes of bounded w…
Complete Reducibility and Separability
设G是特征p > 0的代数闭域上的约化线性代数群.称G的子群在G中是可分的，如果它的整体中心化子和无穷小中心化子具有相同的维数。研究了G的子群的可分性概念与Serre的G-完全约化概念之间的相互作用。可分性假设出现在许多关于G-完全约化的一般定理中。我们证明，许多这些结果失败，没有这个假设。另一方面，我们证明了如果G是一个连通的约化群，且p对G是很好的，则G的任何子群都是可分的;我们推出了在G的这些假设下，如果G的李代数是半单的H-模，则G的子群H是G-完全约化的.最近，Guralnick证明了：如果H是G的一个约化子群，C是G的一个共轭类，则C与H的交是H-共轭类的有限并。对于一般的p -当某些额外的假设成立，包括可分性-这是从一个众所周知的切空间的论点，由于理查森，但在一般情况下，它取决于Lusztig的深刻结果，即一个连接的约化群只有200多个幂单共轭类。我们表明，类似的Guralnick的结果是假的，如果考虑共轭类的n元组的元素从H n > 1。
We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class $\mathfrak{M}$ are are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of t…
Bounding conjugacy depth functions for wreath products of finitely generated abelian groups
本文研究了阿贝尔群圈积的共轭可分性渐近行为，完全刻画了lamplighter群的渐近类并给出了广义lamplighter群及无限基群情形的指数或超指数上下界，进而为非阿贝尔作用群的各种群圈积的共轭深度函数提供了下界。
Let $G$ be a simple algebraic group of type $E_n (n=6,7,8)$ defined over an algebraically closed field $k$ of characteristic $2$. We present examples of triples of closed reductive groups $H<M<G$ such that $H$ is $G$-completely reducible, but not $M$-completely reducible. As an application, we consider a question of Külshammer on representations of finite groups in reductive groups. We also consider a rationality problem for $G$-complete reducibility and a problem concerning conjugacy classes.
In this paper we study conjugacy and subgroup separability properties in the class of nilpotent $\mathbb{Q}[x]$-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over naturally to this more general class. Among other results, we offer a generalization of a theorem due to G. Baumslag. The generalized version states that if $G$ is a finitely $\mathbb{Q}[x]$-generated $\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered group and $H$ is a $\mathbb{Q}[x]$-isolated subgroup of $G,$ then for any prime $π\in \mathbb{Q}[x]…
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $k$. The aim of this paper is to present a method to find triples $(G,M,H)$ with the following three properties. Property 1: $G$ is simple and $k$ has characteristic 2. Property 2: $H$ and $M$ are closed reductive subgroups of $G$ such that $H<M<G$, and $(G,M)$ is a reductive pair. Property 3: $H$ is $G$-completely reducible, but not $M$-completely reducible. We exhibit our method by presenting a new example of such a triple in $G=E_7$. Then we consider a rationality problem and a problem concerning con…
The Zieschang-McCool method for generating algebraic mapping-class groups
设g和p为非负整数。设A（g，p）表示由{t_1，.上的自由群的所有自同构组成的群，t_p，x_1，.，x_g，y_1，. y_g}，其固定元素t_1t_2. t_p[x_1，y_1]. [x_g，y_g]并置换共轭类的集合{[t_1]，.，[t_p]}。Labruère和巴黎，建立在工作的阿廷，马格努斯，德恩，尼尔森，Lickorish，Zieschang，Birman，汉弗莱斯，和其他人，表明，A（g，p）是由一个集，这是所谓的ADLH集。我们使用的方法Zieschang和McCool给一个自足的，代数证明这一结果。Labruère和巴黎也给出了定义关系的ADLH集在A（g，p）;我们不知道一个代数证明这g > 1。考虑亏格为g的可定向曲面S（g，p），有p个穿孔，使得（g，p）不是（0，0）或（0，1）. S（g，p）的代数映射类群记作M（g，p），定义为具有生成集{t_1，.，t_p，x_1，.，x_g，y_1，. y_g}和关系式t_1t_2. t_p[x_1，y_1]. [x_g，y_g]，其置换共轭类的集合{[t_1]，.，[t_p]}。现在从尼尔森的结果得出，M（g，p）是由ADLH集的图像与反射一起生成的。这给出了一种新的方法来证明M（g，p）等于S（g，p）的（拓扑）映射类群，沿着Magnus、Karrass和Solitar在1966年提出的路线。
Twisted conjugacy separable groups
我们研究了扭曲共轭可分性的概念（基本上是在我们以前的文章中介绍的扭曲版本的Burnside-Frobenius定理的证明）和一些相关的性质。我们给出了具有和不具有此属性的群的例子，并在某些扩展下研究了其行为。对多圈有限群的扭曲Dehn共轭问题给出了肯定的回答。指出了今后研究中应注意的问题。
Residual finiteness properties of some of Halls groups
本文研究了某些Halls群的余有限性质，构造了具有不同共轭分离增长和余有限性增长的群类，并首次展示了余有限性增长介于多项式与指数之间的例子。
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