Cohomology of profinite groups of bounded rank

高秩体积的 profiniteness

Profiniteness of higher rank volume

本文证明了高秩半单李群中不可约格的协体积由 profinite 完备化决定，并在无 CSP 条件下证明八元数双曲同余流形的体积是 profinite 不变量。

有界上同调不是profinite不变量

Bounded cohomology is not a profinite invariant

本文证明了有界上同调并非profinite不变量，通过构造具有同构profinite完备化但实二阶有界上同调不同的剩余有限群对，并利用Galois上同调给出了允许此类例子的所有高秩Lie群的完整列表。

On profinite groups in which centralizers have bounded rank

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p group, then G is virtually of rank at most r+1.

On the profinite rigidity of lattices in higher rank Lie groups

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$, $F_4$, and $G_2$. In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\mathrm{SL}_{2n+1}(\mathbb{R})$, $\mathrm{SL}_{2n+1}(\mathbb{C})$, $\mathrm{SL}_n(\mathbb{H})$, or groups of type $E_6$.

Profinite Completion of Free Pro-$\mathcal{C}$ Groups

We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite rank, is a free pro-$\CC$ group as well, of the corresponding rank.

On absolutely profinitely solitary lattices in higher rank Lie groups

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in $\operatorname{Sp}(n,\mathbb{R})$, $G_{2(2)}$, $E_8(\mathbb{C})$, $F_4(\mathbb{C})$, and $G_2(\mathbb{C})$ are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Pontryagin对偶与profinite模层

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。

Groups with the same cohomology as their profinite completions

For any positive integer $n$, $\mathcal{A}_n$ is the class of all groups $G$ such that, for $0\leq i\leq n$, $H^i(\hat{G},A)\cong H^i(G,A)$ for every finite discrete $\hat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes $\mathcal{A}_n$. In addition, we investigate the residually finite groups in the class $\mathcal{A}_2$.

Cohomology and profinite topologies for solvable groups of finite rank

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\to \hat{G}_p$ induces an isomorphism $H^\ast(\hat{G}_p,M)\to H^\ast(G,M)$ for any discrete $\hat{G}_p$-module $M$ of finite $p$-power order. For the general case, we prove that $G$ contains a normal subgroup $N$ of finite index such that the map $H^\ast(\hat{N}_p,M)\to H^\ast(N,M)$ is an isomorphism for any discrete $\hat{N}_p$-module …

profinite 型群与 profinite 刚性

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。

Ascending HNN extensions of polycyclic groups have the same cohomology as their profinite completions

Assume $G$ is a polycyclic group and $ϕ:G\to G$ an endomorphism. Let $G\ast_ϕ$ be the ascending HNN extension of $G$ with respect to $ϕ$; that is, $G\ast_ϕ$ is given by the presentation $$G\ast_ϕ= < G, t \ |\ t^{-1}gt = ϕ(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_ϕ}$ be the profinite completion of $G\ast_ϕ$. We prove that, for any finite discrete $\hat{G\ast_ϕ}$-module $A$, the map $H^*(\hat{G\ast_ϕ}, A)\to H^*(G\ast_ϕ,A)$ induced by the canonical map $G\ast_ϕ\to \hat{G\ast_ϕ}$ is an isomorphism.

The $p$-Zassenhaus Filtration of a Free Profinite Group and Shuffle Relations

For a prime number $p$ and a free profinite group $S$ on the basis $X$, let $S_{(n,p)}$, $n=1,2,\ldots,$ be the $p$-Zassenhaus filtration of $S$. For $p>n$, we give a word-combinatorial description of the cohomology group $H^2(S/S_{(n,p)},\mathbb{Z}/p)$ in terms of the shuffle algebra on $X$. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

On countable coverings of word values in profinite groups

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has…

S-arithmetic spinor groups with the same finite quotients and distinct $\ell^2$-cohomology

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.

Pólya-Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya-Carlson dichotomy between rationality and a natural boundary for the an…

具有自由商的扩张之间缺乏profinite刚性

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。

On profinite groups in which commutators are covered by finitely many subgroups

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…

Profinite non-rigidity of arithmetic groups

We show that for a typical high rank arithmetic lattice $Γ$, there exist finite index subgroups $Γ_{1}$ and $Γ_{2}$ such that $Γ_{1} \not\simeq Γ_{2}$ while $\widehat{Γ_{1}} \simeq \widehat{Γ_{2}}$. But there are exceptions to that rule.

A profinite analogue of Lasserre's theorem

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of finite presentability.

On hereditarily just infinite profinite groups obtained via iterated wreath products

We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.

Profinite groups in which centralizers are virtually procyclic

The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.

Adelic superrigidity and profinitely solitary lattices

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.

On Profinite Groups of Type $\operatorname{FP}_\infty$

Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G \in \widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$ is of type $\operatorname{FP}_\infty$ over $R$, then there is some $n$ such that $H_R^n(G,R [[ G ]]) \neq 0$, and deduce that torsion-free soluble pro-$p$ groups of type $\operatorname{FP}_\infty$ over $\mathbb{Z}_p$ have finite rank, thus answering the torsio…

Free subgroups of finitely generated free profinite groups

We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for …

有限群的坚实对偶性

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。

On verbal subgroups in finite and profinite groups

Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.

Profinite 完备化与上同调跳跃轨迹

Profinite completions and cohomology jump loci

本文证明 profinite 完备化在特定假设下决定 CW 复形的上同调跳跃轨迹及言语子群分次商，并应用于超平面排列，揭示其不区分算术 Zariski 对且不决定排列群。

Baumslag–Solitar 群及其推广的上同调可分性

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。

关于profinite群中可定义集合的测度

On the measure of definable sets in profinite groups

本文研究profinite群中言语集合的Haar测度，通过联系词宽找到正测度非一集合，给出有限秩自由情形下测度为零的充分条件，并计算无挠有限生成阿贝尔群中可定义集合的测度。

Bieri-Eckmann Criteria for Profinite Groups

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…

一致性的 profinite 识别

Profinite recognition of uniformity

该研究针对非双曲等距群的半单李群不可约格，证明有理秩（除 D_{2n} 型外）、一致性（除 D_4 型外）为 profinite 不变量，且 D_4 型中存在一致与非一致的 profinite 同构格。

结群的分支子群及其 profinite 和上同调结构

Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure

本文研究结群的分支子群及其 profinite 和上同调结构，通过定义meridional inertia子群和全局分支子群，分析其在有限商、profinite 完成和上同调中的性质。

Presentations of Finite Simple Groups: Profinite and Cohomological Approaches

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.

2-轨道群的中央扩张的 profinite 刚性性质

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。

关于 S-adic 移位空间的 profinite 方法 I：饱和指令序列

Profinite approach to S-adic shift spaces I: Saturating directive sequences

本文研究了 S-adic 移位空间的 profinite 方法，提出饱和指令序列的概念，并探讨了其与可识别性的关系及 Schützenberger 组的秩性质。

随机与完美 profinite 群中的高秩与多重性

High Rank and Multiplicity in Random and Perfect Profinite Groups

本文证明随机 profinite 群几乎必然有限生成且有限呈现，并构造唯一的万有 d-生成完美 profinite 群，解决相关猜想，方法基于 crown 理论的高秩多重性洞见。

Profinite commensurability of S-arithmetic groups

Given an S-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places S is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of a higher rank S-arithmetic group determines the number field up to arithmetical equivalence and the places in S above unramified primes. We include applications to profiniteness questions of group invariants.

不可数秩的Demushkin群

Demushkin groups of uncountable rank

本文将Demushkin群理论从可数生成情形推广到任意秩，研究其代数性质、同构类、绝对Galois群实现及投射有限完备化。

The lower rank of direct products of hereditarily just infinite groups

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

On the profinite topology of right-angled Artin groups

We give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of the direct product of two free groups. We show that the profinite topology of the above group is strongly connected with the profinite topology of the free group of rank two.

Protoroi的profinite子群格

The Lattice of Profinite Subgroups of Protori

本文研究紧致连通阿贝尔群（protoroi）的profinite子群格，利用分解定理证明其构成同源类格，并构造万有分解，使对偶protorius的核自然涌现，从而支持有限秩无挠群作为子群的研究。

仿射考克斯特群的射有限刚性

Profinite rigidity of affine Coxeter groups

本文证明不可约仿射考克斯特群具有一阶刚性与绝对射有限刚性，其一阶理论无素模型，并证明有限秩泛考克斯特群及双曲单端直角考克斯特群是齐性的。

由子群和元组诱导的度量

Subgroup measures and profinite rigidity in free groups

本文研究了由自由群子群和元组诱导的度量，证明了子群的 profinite 刚性与元组的刚性等价，并推广了相关结果。

On the rank of compact p-adic Lie groups

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G the rank rk(G) is greater than or equal to dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank …

无 torsion 的灯塔群的 profinite 刚性

The Profinite Rigidity of Torsion-Free Lamplighter Groups

该研究证明了无 torsion 灯塔群在绝对意义上是 profinite 刚性的，其有限商唯一确定其同构类型。

Symmetric continuous cohomology of topological groups

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic (J. Algebra 322 (2009), 1360-1378), where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also defi…

The Congruence Subgroup Problem for low rank Free and Free Metabelian groups

The congruence subgroup problem for a finitely generated group $Γ$ asks whether $\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $Γ=F_{2}$ and $Φ_{2}$) and a new result for $Γ=Φ_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(Φ_{2}\right)=\hat{F}_ω$ when $Φ_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_ω$ is the free pr…

Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.

算术格的非有限刚性及Kähler性质

Profinite Non-Rigidity of Arithmetic Lattices and the Kähler Property

本文研究算术格的非有限刚性，刻画了允许非同构实形式的高秩李群绝对Dynkin类型，并证明有限呈现剩余有限群的Kähler性质不由其有限完备化决定。

Cohomology of absolute Galois groups

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-$p$ case, i.e., one would like to know which pro-$p$ groups occur as maximal pro-$p$ Galois groups, i.e., maximal pro-$p$ quotients of absolute Galois groups. Indeed, pro-$p$ groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-$p$ groups, cal…

有限完备化的平凡性问题

The triviality problem for profinite completions

本文证明有限呈示群是否具有非平凡有限商不可判定，并将结果推广到非正曲率方形复形基本群及双曲群的多种性质。

On cohomology rings of infinite groups

Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.

Rank, coclass and cohomology

We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of bounded rank.

Rational discrete first degree cohomology for totally disconnected locally compact groups

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of…