The Figure Eight Knot Group is Conjugacy Separable
S. C. Chagas, P. A. Zalesskii
证明了PGL（2，C）（抽象）可与欧氏Bianchi群同构的无无挠的子群是共轭可分的.因此，我们推导出标题中所述的结果。
We prove that torsion free subgroups of PGL(2,C) (abstractly) commensurable with the Euclidean Bianchi groups are conjugacy separable. As a consequence we deduce the result stated in the title.
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The Fundamental group of a finite graph of conjugacy separable groups with finite edge groups is conjugacy separable
本文证明了具有有限边群的共轭可分群的有限图的基本群是共轭可分的，肯定回答了Ashot Minasyan提出的自然问题。
We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a one-relator group with torsion is also conjugacy separable.
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. It is proved that the fundamental group of a hyperbolic 3-manifold (closed or with cusps) is subgroup conjugacy separable.
We show that the class of $\mathcal{C}$-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class $\mathcal{C}$ is an extension closed variety of finite groups. As a consequence we show that the class of $\mathcal{C}$-conjugacy separable groups is closed under taking arbitrary graph products. In particular, we show that right angled Coxeter groups are hereditarily conjugacy separable and 2-hereditarily conjugacy separable, and we show that infinitely generated right angled Artin groups are hereditarily conjugacy separable and $p$-hereditarily co…
We prove that non-uniform arithmetic lattices of $SL_2(\mathbb{C})$ and in particular the Bianchi groups are conjugacy separable. The proof based on recent deep results of Agol, Long, Reid and Minasyan.
We prove that any word hyperbolic group which is virtually compact special (in the sense of Haglund and Wise) is conjugacy separable. As a consequence we deduce that all word hyperbolic Coxeter groups and many classical small cancellation groups are conjugacy separable. To get the main result we establish a new criterion for showing that elements of prime order are conjugacy distinguished. This criterion is of independent interest; its proof is based on a combination of discrete and profinite (co)homology theories.
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.
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…
Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.
$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products
该研究给出限制 wreath 积为 $\boldsymbol{\textit{C}}$-遗传共轭可分群的充要条件，证明 Grigorchuk 群是 2-遗传共轭可分群，构造了无穷多类非多循环遗传共轭可分群。
We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.
A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable
证明了任何Seifert 3-流形的基本群是共轭可分的。也就是说，共轭类可以在有限同分中区分，或者等价地，共轭类在pro-finite拓扑中是封闭的。
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。
Free-by-cyclic groups are conjugacy separable
本文证明了所有有限生成的自由-循环群都是共轭可分离的，并展示了外自同构群的残余有限性。
We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, $C'(1/6)$ one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy…
We prove that every subnormal subgroup of p-power index in a right-angled Artin group is conjugacy p-separable. As an application, we prove that every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups. As another application, we prove that the outer automorphism group of a right-angled Artin group is virtually residually p-finite. We also prove that the Torelli group of a right-angled group is residually torsion-free nilpotent, hence residually p-finite and bi-orderable.
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 the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this property and study its behavior under some extensions. An affirmative answer to the twisted Dehn conjugacy problem for polycyclic-by-finite group is obtained. Some problems for the further study are indicated.
It is proved that generalized free product of two finite p-groups is a conjugacy p-separable group if and only if it is residually finite p-groups. This result is then applied to establish some sufficient conditions for conjugacy p-separability of generalized free product of infinite groups.
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 …
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。
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 $G$. We prove that free groups and the fundamental groups of orientable closed compact surfaces are SCS.
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.
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。
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。
Let $π$ be a set of primes. We show that $π$-separable groups have a conjugacy class of $\mathfrak F$-injectors for suitable Fitting classes $\mathfrak F$, which coincide with the usual ones when specializing to soluble 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. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.
Let $π$ be a set of primes. We show that $π$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.
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。
Let M = H^3 / Γbe a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γand g is in Γ, then the double coset HgK is separable in Γ. As a consequence we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of M is conjugacy separable then so is the fundamental group of M. Invoking recent work of Agol and Wise, it follows that if M is a compact, orientable 3-manifold then π_1(M) is conjugacy separable.
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。
We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.
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。
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。
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…
On conjugacy and perturbation of subalgebras
本文研究迹冯·诺依曼代数中子代数的共轭轨道，构造了使具有Haagerup性质或性质(T)的子代数分别具有唯一嵌入的II$_1$因子，并讨论了可分性与密度特征。
Groups with conjugacy classes of coprime sizes
针对有限群中位于互质大小共轭类的元素，证明其共轭类生成子群的交为阿贝尔正规子群，并推广$π$-正则元共轭类乘积的性质，扩展了$p$-正则共轭类公因子图的相关结果。
In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinct pair of conjugacy classes.
Residual $p$ properties of mapping class groups and surface groups
令$\mathcal M (Σ, \mathcal P)$是穿孔定向表面$(Σ, \mathcal P)$的映射类组（其中$\mathcal P$可以为空），并且令$\mathcal T_p(Σ,\mathcal P)$是$\mathcal M (Σ, \mathcal P)$对$H_1 (Σ\setminus \mathcal P, \mathbb F_p)$的动作的内核。证明了$\mathcal T_p(Σ, \mathcal P)$是剩余$p$ .特别地，这表明$\mathcal M (Σ, \mathcal P)$实际上是剩余的$p$。对于群$G$，我们用$\mathcal I_p(G)$表示${\rm Out} (G)$对$H_1(G, \mathbb F_p)$的自然作用的核。为了实现我们的定理，我们证明了，在一定的条件下（$G$是共轭$p$ -可分的，并具有性质A），该集团$\mathcal I_p(G)$是剩余$p$。自由基团和表面基团具有性质A的事实是由于格罗斯曼。自由群是共轭$p$ -可分的这一事实是由Lyndon和Schupp提出的.事实上，表面群是共轭$p$ -可分的，从技术的角度来看，主要结果的文件。
Groups with triangle-free graphs on p-regular classes
该研究针对素数p，对满足“任三个非中心p-正则元共轭类中必有两类长度互素”的有限p-可分群的p-结构展开分类，推进了群论中相关结构问题的研究。
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。
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。
A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …
Groups whose common divisor graph on $p$-regular classes has diameter three
本文针对有限p-可分群的p-正则元素非中心共轭类公因子图Γ_p(G)，正面解答了最大基数顶点与其他顶点最大距离的公开问题，并研究了该图直径为3时群的p-结构。
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy proble…
Aritmethic lattices of $\SO(1,n)$ and units of group rings
本文证明 $\mathrm{SO}(1,n)$ 标准算术子群的共轭可分性，并将其用于推导特定整数群环单位群的同一性质及有限商群对原群环的可确定性。
On the residual finiteness of outer automorphisms of relatively hyperbolic groups
证明了共轭可分双曲群的外自同构群的每个虚扭自由子群是剩余有限的。由此，我们能够证明：每一个可生成的富克斯群和每一个自由有限群的外自同构群都是剩余有限的.对于相对双曲群，我们也推广了主要结果。
Subgroups of the direct product of graphs of groups with free abelian vertex groups
本文将 Baumslag–Roseblade 关于自由群直积有限展示子群的结果推广到具有自由阿贝尔顶点群与循环边群的群图，并证明相关 RAAG 子群的算法问题可判定。
Let $H$ be a torsion-free $δ$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $δ$, $\sharp{S}$ and $\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\mathcal{C}=\mathcal{C}(δ, \sharp S)$ such that, for any endomorphism…
Cocycle and Orbit Equivalence Superrigidity for Malleable Actions of w-Rigid Groups
证明了：如果可数离散群$Γ$是{ \it w-刚性}，即它包含一个具有相对性质（T）的无穷正规子群$H$（例如$Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$，或$Γ= H \times H'$，其中$H$是无限Kazhdan群，$H'$是任意的），并且$\Cal V$是有限冯诺依曼代数的酉群的闭子群（例如，$\Cal V$可计数离散或可分离紧凑），则概率空间$(X,μ)$上$Γ$的保测度作用$Γ\curvearrowright X$的任何$\Cal V$ -值可测上循环是$H$上的弱混合且{ \it s-可延展}（例如伯努利作用$Γ\curvearrowright [0,1]^Γ$）与$Γ$到$\Cal V$的群态射共同源。我们使用的情况下$\Cal V$离散的这个结果证明，如果此外$Γ$没有非平凡的有限正规子群，然后任何轨道之间的等价$Γ\curvearrowright X$和自由遍历措施保持行动的可数组$Λ$是实现了共轭的行动，关于一些组同构$Γ\simeq Λ$。
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