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. Let G be a profinite group in which each element has either finite or prime power (possibly infinite) order. Then G is either torsion or virtually pro-p for some prime p.
In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.
A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove that if a profinite group $G$ admits a coprime automorphism $φ$ of prime order such that every fixed point of $φ$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup. A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ a…
We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally if $\mathbf H$ is variety of finite groups closed under extension and containing $\mathbb Z/p\mathbb Z$ for infinitely may primes $p$, the corresponding result holds for free pro-$\bar{\mathbf H}$ monoids.
We prove every regular element of a free profinite monoid generates a prime ideal; in particular the minimal ideal is prime. The latter result was first proved by Almeida and Volkov using techniques from symbolic dynamics; our proof is elementary.
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.
It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to ${\mathbb Z}_p$.) A weaker result is proved for $p=2$.
An element $g$ of a group $G$ is said to be right Engel if for every $x\in G$ there is a number $n=n(g,x)$ such that $[g,{}_{n}x]=1$. We prove that if a profinite group $G$ admits a coprime automorphism $φ$ of prime order such that every fixed point of $φ$ is a right Engel element, then $G$ is locally nilpotent.
Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g.$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?
The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.
The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic for each nontrivial element a in A. Then G' is locally finite.
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. Given a set of primes $π$, we take interest in profinite groups with restricted centralizers of $π$-elements. It is shown that such a profinite group has an open subgroup of the form $P\times Q$, where $P$ is an abelian pro-$π$ subgroup and $Q$ is a pro-$π'$ subgroup. This significantly strengthens a result from our earlier paper.
Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $γ_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $γ_k(G)$ is locally virtually nilpotent.
We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hatπ : G \to \hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime field and if $V$ is a $k$-vector space, then, there is a natural isomorphism between $\hat{V}$ (for the underlying additive group structure) and the additive group of the double-dual $V^{**}$. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case.
We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.
We introduce the notion of corestricted free products of a family of profinite groups indexed over an arbitrary profinite space. Using arithmetic results of the second author, this enables us to prove an analogue of Riemann's existence theorem for the decomposition groups of certain infinite sets of primes of a number field.
We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup…
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…
In this paper we investigate the following general problem. Let $G$ be a group and let $i(G)$ be a property of $G$. Is there an integer $d$ such that $G$ contains a $d$-generated subgroup $H$ with $i(H)=i(G)$? Here we consider the case where $G$ is a profinite group and $H$ is a closed subgroup, extending earlier work of Lucchini and others on finite groups. For example, we prove that $d=3$ if $i(G)$ is the prime graph of $G$, which is best possible, and we show that $d=2$ if $i(G)$ is the exponent of a finitely generated prosupersolvable group $G$.
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…
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.
Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$ is of order $q^3$ we prove that if some power of each element in $C_G(a)$ is Engel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Some analogues consequences of quantitative nature for finite groups are also obtained.
Let $G$ be 2-generated group. The generating graph $Γ(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $Γ(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $Δ(G)$ obtained from $Γ(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinit…
In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.
Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). \\ We define a constant $c_n$ for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying $c_n<1$. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the …
Brauer Theory for a finite group can be viewed as a method for comparing the representations of the group in characteristic 0 with those in prime characteristic. Here we generalize much of the machinery of Brauer theory to the setting of profinite groups. By regarding Grothendieck groups as functors we describe corresponding Grothendieck groups for profinite groups, and generalize the decomposition map, regarded as a natural transformation. We discuss characters and Brauer characters for profinite groups. We give a functorial description of the block theory of a profinite group. We finish with…
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 …
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.
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.
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 $r\geq 3$…
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 …
A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same primes. In the present paper we establish that a profinite group $G$ is finite-by-pronilpotent if the cardinality of the set of coprime commutators in $G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite commutator subgroup $G'$ if the cardinality of the set of anti-coprime commutators in $G$ is less than $2^{\aleph_0}$.
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…
We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite' group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the…
Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\leq m$ for all $g\in G$,…
A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if every subgroup of finite index is open. We show that two profinite groups that are elementarily equivalent, in the first-order language of groups, are isomorphic if one of them is strongly complete, extending a result of Moshe Jarden and Alexander Lubotzky which treats the case of finitely generated profinite groups.
Let $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually c…
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…
Given a polynomial of degree d over a number field, the image of the associated arboreal representation of the absolute Galois group of the field is a profinite group acting on the d-ary tree. Boston and Jones conjectured that for a quadratic polynomial, the image of such a representation contains a dense set of settled elements. Here an element is settled if it exhibits a certain pattern of growth of cycles at finite levels of the tree. In this paper, we prove the conjecture of Boston and Jones generically in the case when the quadratic polynomial has a strictly pre-periodic post-critical orb…
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 …
