Extensions of profinite duality groups
Alexander Schmidt, Kay Wingberg
证明了若核满足一定的有限性条件，则该类有限对偶群在群扩张下是闭的。这推广了Pletch和Wingberg的早期结果。
We show that the class of profinite duality groups is closed under group extensions provided that the kernel satisfies some finiteness condition. This extends earlier results of Pletch and of Wingberg.
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Pontryagin duality and sheaves of profinite modules
本文研究profinite群的Pontryagin对偶理论，填补了在离散模系数上同调研究中缺失的对偶构造这一理论空白。
Solid Duality for Profinite Groups
研究有限群同调与上同调间类似庞加莱对偶性，通过凝聚数学框架分类此类群，统一推广现有结果并构造新例，是利用该框架研究有限群（上）同调的首批工作之一。
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
Profinite genus of HNN-extensions with finite associated subgroups
本文研究了关联子群为有限群的HNN扩张的概型，给出了其同构类数量的公式并探讨了概型确定性的条件。
We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\famS$ and the classes of `relative extensions' of $G$ by abelian groups, modulo a certain equivalence relation. We establish this correspondence for both discrete and profinite group pairs. We go on to discuss the relationships of profinite group pairs of cohomological dimension one with free products and projective group pairs.
In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the …
Projective pairs of profinite groups
我们将投射投射有限群的概念推广到一个投射有限群和一个闭子群的投射对。我们建立了与PAC域的伪代数闭（PAC）扩张的联系：设M是PAC域K的代数扩张.则M/K是PAC当且仅当对应的一对绝对伽罗瓦群（Gal（M），Gal（K））是投射的。此外，任何射影对都可以被实现为PAC域的PAC扩张的绝对伽罗瓦群。使用这个特征，我们构建了相对较小字段的PAC扩展的新示例，例如，有理数的无界阿贝尔扩张。
Groups of profinite type and profinite rigidity
本文研究 profinite 型群（可实现为 Galois 群的群）及其 profinite 刚性（唯一 profinite 拓扑），探讨阿贝尔群与群扩张的性质，并联系上同调比较映射的单射性与满射性。
Lack of profinite rigidity among extensions with free quotient
本文构造了无穷多个互不同构的半直积，它们具有相同的profinite完备化，从而证明在自由商扩张中缺乏profinite刚性，并给出了自由群、自由阿贝尔群和曲面群的具体参数条件。
Profinite rigidity properties of central extensions of 2-orbifold groups
研究2-轨道群中央扩张的profinite刚性性质，证明刚性和非刚性现象并存，非刚性现象在扩展后消失。
On free profinite subgroups of free profinite monoids
我们回答了Margolis从1997年的一个问题，建立了一个自由生成的自由profinite幺半群的极小理想的极大子群是一个自由的投射有限群。更一般地说，如果$\mathbf H$是扩张闭的有限群簇，并且对于无限个素数$p$包含$\mathbb Z/p\mathbb Z$，则相应的结果对于自由pro- $\bar{\mathbf H}$幺半群成立。
In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type $\operatorname{FP}_n$ over a profinite ring $R$, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type $\operatorname{FP}_n$ is closed under extensions, quotients by subgroups of type $\operatorname{FP}_n$, proper amalgamated free products and proper $\operatorname{HNN}$-extensions, for each $n$. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type…
Jordan-Holder Theorem for profinite groups and applications
本文将合成列与合成因子概念推广到投射有限群，证明其Jordan-Hölder定理，并应用于无限投射可解扩张的Galois定理，同时探讨非强完全投射有限群中抽象与拓扑合成因子的关系。
The generalised Fitting subgroup of a profinite group
有限群的广义Fitting子群是由所有幂零或拟单的次正规子群生成的群。这个子群在有限群理论中的重要性源于这样一个事实，即它总是包含自己的中心化器，因此任何有限群都是其广义Fitting子群的自同构自同构群的阿贝尔扩展。我们定义了一类推广这一现象的投射有限群，并探讨了投射有限群的结构的一些后果。
Sofic groups and profinite topology on free groups
给出了弱sofic群（w-sofic群）的定义.我们的定义是sofic群定义的自然扩张，这里用有限群上的一般双不变度量代替了对称群上的Hamming度量.非W-sofic群的存在性等价于自由群上关于有限拓扑的某些猜想。
Profinite properties of residually free groups
该研究在剩余自由群的投射有限性质方向取得进展，证明了素三维流形基本群与$\mathbb{Z}^n$的直积在有限生成剩余自由群中投射有限刚性，还得到相关投射有限不变量并推广了已有引理。
On just infinite pro-p-groups and arithmetically profinite extensions of local fields
野群是特征p的局部域的野自同构的群。本文应用Fontaine-Wintenberger的范数域场论研究了野群的结构。特别地，我们给出了R的一个新的简短证明.卡米纳定理，它说，每一个亲p-群与可数个开放子群是同构的一个闭子群的野生集团。研究了换位子群异常小的野群的闭子群T.将群T看作是p进域的算术有限扩张的伽罗瓦群，我们肯定地回答了关于局部域的深分歧扩张的Coates-Greenberg问题。最后利用子群T证明了在具有特征p的有限剩余域的交换完备局部Noether整环上的野群不是解析的。
Ascending HNN extensions of polycyclic groups have the same cohomology as their profinite completions
假设$G$是一个循环群，而$ϕ:G\to G$是一个自同态。令$G\ast_ϕ$是$G$关于$ϕ$的HNN升序扩展;即，$G\ast_ϕ$由表示$$G\ast_ϕ= < G, t \ |\ t^{-1}gt = ϕ(g)\ \{for all}\ g\in G >.$$给出。此外，令$\hat{G\ast_ϕ}$是$G\ast_ϕ$的profinite完成。证明了对任何有限离散$\hat{G\ast_ϕ}$ -模$A$，由正则映射$G\ast_ϕ\to \hat{G\ast_ϕ}$诱导的映射$H^*(\hat{G\ast_ϕ}, A)\to H^*(G\ast_ϕ,A)$是同构.
This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $PFP_1$ is closed under extensions. In addition, we prove that…
Normal Subgroups of Profinite Groups of Non-negative Deficiency
本文首次研究了非负亏的投射有限群。本文的主要目的是证明在非负亏的profinite群$G$中存在无穷指数的n-生成正规子群对$G$的结构给出了相当强的推论.为了使这一点更精确，我们引入了profinite群$G$的$p$ -亏格（$p$是素数）的概念。这个概念在研究投射有限群时比亏数的概念更有用。证明了若$G$的$p$ -亏是正的，且$N$是一个可生成的正规子群，使得$G/N$的$p$ -Sylow子群是无穷的，且$p$整除$N$的阶，则上同调$p$ -维数有$\cd_p(G)=2$，$\cd_p(N)=1$和$\vcd_p(G/N)=1$;此外，或者$G/N$的$p$-Sylow子群实际上是环状的，或者$N$的$p$-Sylow子群是环状的。一个在素数$p$（$PD^3$ -群）上的3维Profinite Poincaré对偶群$G$有亏数0。在这种情况下，我们表明，对于上述$N$和$p$，或者$N$在$p$处是$PD^1$，而$G/N$在$p$处实际上是$PD^2$，或者$N$在$p$处是$PD^2$，而$G/N$在$p$处实际上是$PD^1$。特别地，如果$G$是pro-$p$，则$N$是无限循环的并且$G/N$实际上是Demushkin，或者$N$是Demushkin并且$G/N$实际上是无限循环的。我们应用这一结果推导出结构信息的profinite完成的上升HNN-扩展的…
Profinite rigidity of fibring
本文研究了有限呈现的LERF群的代数纤维化性质，证明其为profinite属性，并探讨了极限群乘积的纤维化及庞加莱对偶群的profinite刚性。
Permanence criteria for semi-free profinite groups
引入了投射有限群是半自由群的条件，它比自由群更一般，比拟自由群更严格。特别地，每个投射半自由投射有限群都是自由的。我们证明了自由群通常的持久性可以推广到半自由群。利用这一点，我们得出结论，如果k是可分闭域，则k（（x，y））的许多域扩张具有自由绝对伽罗瓦群。
Profinite groups with a cyclotomic $p$-orientation are introduced and studied. The special interest in this class of groups arises from the fact that any absolute Galois group $G_{K}$ of a field $K$ is indeed a profinite group with a cyclotomic $p$-orientation $θ_{K,p}\colon G_{K}\to\mathbb{Z}_p^\times$ which is even Bloch-Kato. The same is true for its maximal pro-$p$ quotient $G_{K}(p)$ provided the field $K$ contains a primitive $p^{th}$-root of unity. The class of cyclotomically $p$-oriented profinite groups (resp. pro-$p$ groups) which are Bloch-Kato is closed with respect to inverse limi…
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solution…
Groups with the same cohomology as their profinite completions
对于任何正整数$n$，$\mathcal{A}_n$是所有群$G$的类，使得对于$0\leq i\leq n$，对于每个有限离散$\hat{G}$ -模$A$，有$H^i(\hat{G},A)\cong H^i(G,A)$。我们描述了某些类型的自由积与汞合金和HNN的扩展是在某些类$\mathcal{A}_n$。此外，我们还研究了类$\mathcal{A}_2$中的剩余有限群。
On Bohr compactifications and profinite completions of group extensions
本文精确描述了局部紧群半直积的Bohr紧化与profinite完备化结构，给出了交换正规子群情形下的显式刻画，并应用于圈积群，计算了灯夫群和海森堡群的相应对象。
Families preserving isomorphisms via techniques in anabelian geometry: with an application to a generalized Neukirch-Uchida theorem
本文从算术代数几何视角推广保持自同构的族的概念，证明一大类profinit群的相关同构族源自内自同构，并将其应用得到ℓ-拟数域的广义Neukirch-Uchida定理。
Norm rigidity for arithmetic and profinite groups
本文证明具有有界初等生成的 ${\rm{SL}}_n(A)$ 的共轭不变范数具离散或预紧二分性，并将其与算术群及profinite群的刚性结果相关联，推导出对非剩余有限中心扩张近似的约束。
Algebraic extensions in free groups
本文的目的是统一最近三篇独立论文（Ventura 1997、Margolis、Sapir和Weil 2001以及Kapovich和Miasnikov 2002）的观点，其中给出了Takahasi 1951年定理的类似现代版本。我们开发了一个理论的代数扩张自由群，突出的类比和差异，相对于相应的经典领域理论的概念，我们详细讨论的概念，代数闭包。我们应用这一理论的研究和计算的某些代数性质的子群（例如异常，纯，惰性或压缩，被关闭在某些profinite拓扑）和相应的封闭运营商。我们还分析了封闭的子群下的某些方程组的解决方案。
Duality and Local Group Cohomology
最近，Meierfrankenfeld发表了三个定理上同调的有限模。他们涵盖了当地的决定完全约化;当地分裂组扩展;和代表性的当地分裂扩展的双重对偶。在本说明中，我们得到所有三个相结合的同调和上同调的连续性之间的一定的对偶。
On profinite rigidity, Grothendieck pairs, and the second homology of some $3$-orbifold groups
本文开发了一种有效计算双曲有理同调$3$-球面orbifold群第二同调群的方法，并应用于Weeks流形及其相关格点，证明了这些格点的绝对profinite刚性，并构造了Grothendieck对。
We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic tree…
A cohomological translation of the Kaplansky radical for profinite groups
引入任意profinite群和素数p的Kaplansky根的上同调类比，证明对于域，群论版本的H-猜想等价于原猜想，并建立其对于多种域类的成立性。
On a local property of infinite Galois extensions implying the Northcott property
本文证明 Bombieri–Zannier 提出的蕴含 Northcott 性质的局部性质在有限扩张下不被保持，且不能从无穷伽罗瓦扩张的伽罗瓦群上读出，并构造了相应反例。
A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formu…
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