On profinite groups in which centralizers have bounded rank

Commutators, centralizers, and strong conciseness in profinite groups

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form $[x_1,\dots,x_k]$, where $π(x_1)=π(x_2)=\dots=π(x_k)$. Here $π(x)$ denotes the set of prime divisors of the order of $x\in G$. It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that $γ_k(G)$ is f…

具有限制幂中心化的 profinite 群

Profinite groups with restricted centralizers of powers

研究具有有限指数或有限中心化的 profinite 群，证明其存在开正规子群使得商群具有有限指数。

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $γ_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $γ_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$…

中心化子为阿贝尔群的紧致 $p$-adic 解析群

Compact $p$-adic analytic groups in which centralizers are abelian

本文利用多种代数与表示论方法构造了每个非平凡元素中心化子均为阿贝尔群的 pro-$p$ 群，回答了相关公开问题。

Profinite groups in which many elements have prime power order

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-groups yields a detailed description of profinite groups with that property. Then we deal with two generalizations of profinite CP-groups. Theorem 1.2. A profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. L…

Cohomology of profinite groups of bounded rank

We generalise to profinite groups some of our previous results on the cohomology of pro-p groups of bounded sectional p-rank.

On profinite groups with automorphisms whose fixed points have countable Engel sinks

An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is proved that if a profinite group $G$ admits an elementary abelian group of automorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each $a\in A\setminus\{1\}$ every element of the centralizer $C_G(a)$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ i…

On profinite groups with positive rank gradient

We prove that a profinite group $G$ with positive rank gradient does not satisfy a group law. In the case when $G$ is a pro-$p$ group we show that $G$ contains a nonabelian dense free subgroup.

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。

好群的有限完备化中的有限子群

Finite subgroups of the profinite completion of good groups

本文证明剩余有限好群的有限 $p$-子群共轭类可自然双射对应到其有限完备化，并刻画中心化子与正规化子的闭包性质，且推广至有限可解子群并给出几何群论应用。

A restricted Magnus property for profinite surface groups

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are …

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.

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…

Lannes的T-函子与 profinite 群的 mod-p 上同调

Lannes' $T$-functor and mod-$p$ cohomology of profinite groups

本文通过发展profinite空间参数化的离散扭模族乘积理论，证明了Lannes-Quillen定理对所有profinite群的完整版本，并应用于p- torsion元素和有限p-子群的共轭分离问题。

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.

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 …

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.

On groups with automorphisms whose fixed points are Engel

We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite gr…

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 …

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。

Centralizers of reflections and reflection-independence of Coxeter groups

A Coxeter group W is called reflection independent if its reflections are uniquely determined by W only, independently on the choice of the generating set. We give a new sufficient condition for the reflection independence, and examine this condition for Coxeter groups in certain classes, possibly of infinite ranks. We also determine the finite irreducible components of another Coxeter group, that is a subgroup of W generated by the reflections centralizing a given generator of W. Determining such a subgroup makes our criterion efficient.

Homogeneous number of free generators

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group Γwhich admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group Γis uncountable.

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。

中心化子过剩作为卡尔森深度猜想的一个阻碍

Centralizer Excess as an Obstruction to Carlson's Depth Conjecture

该研究以SmallGroup(128,859)为反例，揭示卡尔森深度猜想失效的机制，提出中心化子过剩准则，证明该反例的直积仍为反例，并分析了相关上同调类的低次表现。

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.

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms $\mathcal{A}_m$ of an $m$-ary tree $\mathcal{T}_m$ in \cite{BS}, to the general case where the permutation group induced on the first level of the tree has $s\geq 1$ orbits. We prove that such a group $A$ embeds in a self-similar abelian group $A^*$ which is also a maximal abelian subgroup of $\mathcal{A}_m$. The construction of $A^*$ is based on the definition of a free monoid $Δ$ of rank $s$ of partial diagonal monomorphisms of $\mathcal{A}_m$, which is used to determine the structure of $C_{\mathcal…

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。

On centralizers of parabolic subgroups in Coxeter groups

Let $W$ be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer $Z_W(W_I)$ of an arbitrary parabolic subgroup $W_I$ into the center of $W_I$, a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. Moreover, by using our description of $Z_W(W_I)$, we reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.

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。

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.

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。

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。

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 …

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.

Centralizers in the Group of Interval Exchange Transformations

We study the group of interval exchange transformations. Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged subintervals. We prove that if $T$ satisfies Keane's infinite distinct orbit condition and $\text{rank}(T)>1+\lfloor m/2 \rfloor$ then the only interval exchange transformations which commute with $T$ are its powers. In the case that $T$ is a minimal 3-interval exchange transformation, we prove a more precise result: $T$ has a trivial centralizer in the group of interval exc…

The centralizers of root subgroups in Kac-Moody Steinberg groups

For the affine and hyperbolic root system the symmetric part of the centralizers of root subgroups in the corresponding Steinberg groups are calculated. In the affine case the corresponding root subsystems can be computed in term of the centralizers in the spherical root systems, while in the hyperbolic case there emerges a "zoo" of examples, many of them non-hyperbolic. This also delivers many examples of naturally occuring root subsystems of infinite rank.

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。

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…

Centralizer of the elementary subgroup of an isotropic reductive group

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that, for any maximal ideal M of R, the rank of the relative root system of G_{R_M} is greater or equal than 2. We show that under this assumption the centralizer of E(R) in G(R) coincides with the abstract group-theoretic center of G(R) and with Cent(G)(R). This generalizes a result of E. Abe and J. Hurley.

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。

The isomorphism type of the centralizer of an element in a Lie group

Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.