Nikolay Nikolov
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.
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We generalise to profinite groups some of our previous results on the cohomology of pro-p groups of bounded sectional p-rank.
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。
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.
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 $E_6$.
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.
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.
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…
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…
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
对于一族群字$w$，我们证明了：如果$G$是一个profinite群，其中所有的$w$ -值都包含在一个具有规定性质的多个子群的并集中，则$w(G)$也具有相同的性质.特别是，我们证明了这一点的情况下，子群是定期或有限秩。若$G$包含有限指数$e$的2/3子群$G_1,G_2,...,G_s$，且其并集包含$G$中的所有$γ_k$ -值，则证明了$γ_k(G)$具有有限的$(e,k,s)$ -有界指数.若$G$包含有限秩$r$的2/3子群$G_1,G_2,...,G_s$，且其并集包含所有$γ_k$ -值，则证明了$γ_k(G)$有有限$(k,r,s)$ -有界秩.
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 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.
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.
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.
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.
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…
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 …
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.
Cohomology and profinite topologies for solvable groups of finite rank
设$G$是一个可解群，其初等交换截面都是有限的.进一步假设$p$是素数，使得$G$不包含任何同构于$C_{p^\infty}$的子群。我们证明了如果$G$是幂零的，那么对于任何有限$p$ -幂阶的离散$\hat{G}_p$ -模$M$，pro-$p$完备映射$G\to \hat{G}_p$诱导同构$H^\ast(\hat{G}_p,M)\to H^\ast(G,M)$.对于一般情形，我们证明了$G$包含一个有限指数的正规子群$N$，使得映射$H^\ast(\hat{N}_p,M)\to H^\ast(N,M)$同构于任何有限$p$ -幂阶的离散$\hat{N}_p$ -模$M$.此外，如果$G$缺少任何$C_{p^\infty}$ -部分，则子群$N$相对于其亲$p$拓扑享有一些额外的特殊性质。
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。
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…
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。
Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.
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。
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。
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。
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.
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。
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。
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
给出了直角Artin群的图是否子群可分的充要条件。此外，我们还研究了两个自由群的直积的profinite拓扑。我们证明了上述群的profinite拓扑与秩为2的自由群的profinite拓扑是强连通的。
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。
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。
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
一个投射有限群G的秩rk（G）是d（H）的上确界，其中H是G的所有闭子群的集合，d（H）表示H的拓扑生成集的最小基数.一个紧拓扑群G允许一个p-adic李群的结构当且仅当它包含一个有限秩的开pro-p子群。对于每个紧p-adic李群G，秩rk（G）大于或等于dim（G），其中dim（G）表示G作为p-adic流形的维数.本文考虑逆问题：rk（G）的dim（G）定界.有限秩的每一个投射有限群G都有一个极大有限正规子群，即它的周期根。我们的主要结果之一如下。设G是具有平凡周期根的紧p-adic李群，且p是奇数.若G有平凡（p-1）-挠，则rk（G）= dim（G）.
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。
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。
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。
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…
In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills the omission; it gives an amended and abridged version of my original proof.
We study a family of hereditarily just infinite profinite groups obtained by iterated wreath products introduced by J. Wilson in 2010. We find explicit generators for this family in a number of cases using combinatorial methods. We then discuss determination of the lower rank for this family. Finally we examine whether the groups in this family admit a finite profinite presentation.
On profinite rigidity amongst free-by-cyclic groups I: the generic case
本文研究了自由-循环群的 profinite 刚性，证明了在特定条件下，超几何性、纤维秩和特征多项式等是 profinite 完成的不变量，并展示了一般自由-循环群的几乎刚性性质。
Elementary totally disconnected, locally compact groups of higher complexity
本文研究由profinite和离散群通过初等运算组装的第二可数群类，定义了作用于$\aleph_0$-正则树的群，并首次构造出分解秩为$ω^ω+1$的群。
Let $G$ be a group. An element $g \in G$ is called a test element of $G$ if for every endomorphism $φ:G \to G$, $φ(g)=g$ implies that $φ$ is an automorphism. We prove that for a finitely generated profinite group $G$, $g \in G$ is a test element of $G$ if and only if it is not contained in a proper retract of $G$. Using this result we prove that an endomorphism of a free pro-$p$ group of finite rank which preserves an automorphic orbit of a non-trivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-$p$ groups and Demushkin groups. By relating t…
Virtual homology of limit groups and profinite rigidity of direct products
本文利用虚拟 Betti 数刻画与 pro-$p$ 群秩梯度技术，证明自由群和曲面群直积在有限展示剩余自由群中的有限完备刚性，并推出相关猜想与流形结论。
A group $G$ is invariably generated by a subset $S$ of $G$ if $G= s^{g(s)} \mid s\in S$ for each choice of $g(s) \in G$, $s \in S$. Answering two questions posed by Kantor, Lubotzky and Shalev, we prove that the free prosoluble group of rank $d \ge 2$ cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank $d$ and derived length $l$ is invariably generated by precisely $l(d-1)+1$ elements.
Finite axiomatizability of the rank and the dimension of a pro-$π$ group
本文证明了在群的一阶语言中存在单一语句，能够精确识别有界秩pro-π群的秩及其Sylow子群的秩和维度。
In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J-classes of free profinite semigroups. In this paper, the Schützenberger groups of such J-classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic minimal subshifts associated with substitutions. It entails that it is decidable whether a finite group is a quotient of such a profinite group. As a further application, the Schützenberger group of the J-cla…
We show that given a finitely generated LERF group $G$ with positive rank gradient, and finitely generated subgroups $A,B \leq G$ of infinite index, one can find a finite index subgroup $B_0$ of $B$ such that $[G : \langle A \cup B_0 \rangle] = \infty$. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover $G$. We construct a transitive virtually faithful action of $G$ such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite gro…
Protori and Torsion-Free Abelian Groups
本文利用紧致阿贝尔群分解定理证明 protorus 的 profinite 子群构成格，引入非阿基米德维数作为分类不变量，并研究其与无挠阿贝尔群的联系。
The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
设k是一个整体域，k_v是k关于v的完备化，k的非阿基米德位置。设\mathbf{G}是k上的连通单连通代数群，它是k_v-秩1的绝对几乎单的.设G= \mathbf{G}（k_v）.设Γ是G中的一个算术格，C=C（Γ）是它的同余核. Lubotzky证明了C是无限的，证实了塞尔的一个早期猜想。本文通过确定C的结构，给出了\Gamm $的同余子群问题的完全解.证明了C是一个自由profinite积，其因子之一是可数生成元上的自由投射有限群\hat{F}_ω.从我们的结果中最令人惊讶的结论是，C的结构只取决于k的特征。C的结构在一些特殊情况下是已知的。其中最重要的例子可能是（非均匀）例子Γ=SL_2（\mathcal{O}（S）），其中\mathcal{O}（S）是k中的S-整数环，S=\{v\}，它在德林费尔德模理论中起着核心作用。证明利用了Lubotzky的分解定理，该定理产生于Γ在与G相关联的Bruhat-Tits树上的作用。
We calculate the rank gradient and p-gradient of free products, free products with amalgamation over an amenable subgroup, and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For the p-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used.
We introduce the notion of a probabilistic identity of a residually finite group. We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable. As an application, we prove a probabilistic variant of the Tits alternative: let G be the profinite completion of a finitely generated linear group. Then either G is virtually solvable, or for any positive integer n, with probability one, n independent, uniformly distributed elements of G freely generate a free subgroup of G of rank n.
Rank gradient and torsion groups
构造了具有正秩梯度的无穷余生成剩余有限扭群的第一个例子。特别是，这些群体是不服从的。讨论了它在成本和$L^2$ -贝蒂数问题中的应用.
We prove that the rank gradient vanishes for mapping class groups of genus bigger than 1, $Aut(F_n)$, for all $n$, $Out(F_n)$ for $n \geq 3$, and any Artin group whose underlying graph is connected. These groups have fixed price 1. We compute the rank gradient and verify that it is equal to the first $L^2$-Betti number for some classes of Coxeter groups.
It is an immediate consequence of the results by Yiftach and Schlage-Puchta that a presentation with p-deficiency greater than one defines a group with positive rank gradient. By results of Button and Thillaisundaram, a finite presentation with p-deficiency greater than one defines a p-large group. Both Yiftach/Schlage-Puchta and Thillaisundaram extended these results to the p-deficiency one case. In this paper we consider the case when the presentation has p-deficiency less than or equal to one.
The rank gradient from a combinatorial viewpoint
本文研究了剩余有限群中有限指数子群的生成元的最小数目的渐近性态。我们分析了三类自然群：顺从群，具有无限可解正规子群的群和虚拟自由群。作为一种工具，我们推广了Lackenby的可分性定理的可分性的情况下，提出了群。
For an arbitrary sequence $(G_s)$ of subgroups of finite index in the generalised Thompson group $$F_{n, \infty} = \langle x_0, x_1, \ldots, x_m, \ldots \mid x_i^{x_j} = x_{i+ n-1} \hbox{ for } i > j \geq 0 \rangle$$ it is shown that $\sup_{s \geq 1} d(G_s) < \infty$ and that the deficiency gradient of $F_{n, \infty}$ with respect to $(G_s)$ is 0 provided $[G : G_s]$ tends to infinity. A higher dimensional analogue is considered for $n = 2$.
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happen…
A $p$-group with positive Rank Gradient
我们为$d\geq 2$和$ε>0$构造了一个由$d$生成的$p$ -群$Γ$，它在渐近意义上的行为几乎类似于由$d$生成的自由亲$p$ -群.我们证明了指数$p^n$的子群需要$(d-ε)p^n$生成元，并且$Γ$的子群增长满足$s_{p^n}(Γ)>s_{p^n}(F_d^p)^{1-ε}$，其中$F_d^p$是$d$生成的自由亲$p$ -群.为了做到这一点，我们引入了一个新的不变量，并研究了它的一些基本性质。
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