Free Profinite Group Research Articles

Mixed moments and the joint distribution of random groups

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random p-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random p-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups.

THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

AbstractFor a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ 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 representations of [formula omitted], [formula omitted] and [formula omitted

By work of Belyĭ [2], the absolute Galois group GQ=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ2), the automorphism group of the free profinite group Fˆ2 on two generators. The image of GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of GTˆ. We do this virtually, namely by showing that a rich class of arithmetically defined representations of GQ can be extended to finite index subgroups of GTˆ. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A=Aut(Fˆ2). We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group Aut(Fd).

The congruence subgroup problem for the free metabelian group on $n\geq4$ generators

The congruence subgroup problem for a finitely generated group \Gamma asks whether the map \widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma}) is injective, or more generally, what is its kernel C(\Gamma) ? Here \widehat{X} denotes the profinite completion of X . It is well known that for finitely generated free abelian groups C(\mathbb{Z}^{n})=\{ 1\} for every n\geq3 , but C(\mathbb{Z}^{2})=\widehat{F}_{\omega} , where \widehat{F}_{\omega} is the free profinite group on countably many generators. Considering \Phi_{n} , the free metabelian group on n generators, it was also proven that C(\Phi_{2})=\widehat{F}_{\omega} and C(\Phi_{3})\supseteq\widehat{F}_{\omega} . In this paper we prove that C(\Phi_{n}) for n\geq4 is abelian. So, while the dichotomy in the abelian case is between n=2 and n\geq3 , in the metabelian case it is between n=2,3 and n\geq4 .

Θ-Hilbertianity

We define Θ-Hilbertianity which generalizes Hilbertianity and show that the absolute Galois group of a countable Θ-Hilbertian PAC field is an appropriate analogue of a free profinite group, thus generalizing a result of Fried-Völklein-Pop.

Free Proalgebraic Groups

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.

The lower p-central series of a free profinite group and the Shuffle algebra

For a prime number p and a free profinite group S on the basis X, let S(n,p), n=1,2,…, be the lower p-central filtration of S. For p>n, we give a combinatorial description of H2(S/S(n,p),Z/p) in terms of the Shuffle algebra on X.

DECIDABLE ALGEBRAIC FIELDS

AbstractWe discuss the connection between decidability of a theory of a large algebraic extensions of ${\Bbb Q}$ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ${\Bbb Q}$ has a decidable existential theory, then within any fixed algebraic closure $\widetilde{\Bbb Q}$ of ${\Bbb Q}$, the field K must be conjugate over ${\Bbb Q}$ to a field which is recursive as a subset of the algebraic closure. We also show that for each positive integer e there are infinitely many e-tuples $\sigma \in {\text{Gal}}\left( {\Bbb Q} \right)^e $ such that the field $\widetilde{\Bbb Q}\left( \sigma \right)$ is primitive recursive in $\widetilde{\Bbb Q}$ and its elementary theory is primitive recursively decidable. Moreover, $\widetilde{\Bbb Q}\left( \sigma \right)$ is PAC and ${\text{Gal}}\left( {\widetilde{\Bbb Q}\left( \sigma \right)} \right)$ is isomorphic to the free profinite group on e generators.

The congruence subgroup problem for low rank free and free metabelian groups

The congruence subgroup problem for a finitely generated group Γ asks whether Aut(Γ)ˆ→Aut(Γˆ) is injective, or more generally, what is its kernel C(Γ)? Here Xˆ denotes the profinite completion of X.In this paper we first give two new short proofs of two known results (for Γ=F2 and Φ2) and a new result for Γ=Φ3:(1)C(F2)={e} when F2 is the free group on two generators.(2)C(Φ2)=Fˆω when Φn is the free metabelian group on n generators, and Fˆω is the free profinite group on ℵ0 generators.(3)C(Φ3) contains Fˆω. Results (2) and (3) should be contrasted with an upcoming result of the first author showing that C(Φn) is abelian for n≥4.

The Cohomology of Canonical Quotients of Free Groups and Lyndon Words

For a prime number p and a free profinite group S , let S^{(n,p)} be the n th term of its lower p -central filtration, and S^{[n,p]} the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group H^2(S^{[n,p]}, \Bbb Z/p) , which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of S^{(n,p)}/S^{(n+1,p)} . We prove that the cohomology group satisfies shuffle relations, which for small values of n fully describe it.

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 subgroups to be free profinite. As a result of our work, we are able to address a conjecture of Jarden on the Hilbertianity of fields generated by torsion points of abelian varieties.

The Zassenhaus filtration, Massey products, and representations of profinite groups

We consider the p-Zassenhaus filtration (Gn) of a profinite group G. Suppose that G=S/N for a free profinite group S and a normal subgroup N of S contained in Sn. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p-cohomological dimension ≤ 1), we prove that Gn+1 is the intersection of all kernels of upper-triangular unipotent (n+1)-dimensional representations of G over Fp. This extends earlier results by Mináč, Spira, and the author on the structure of absolute Galois groups of fields.

Erratum to: Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let k be an algebraically closed field of characteristic p > 0 of countable cardinality. Denote by C a smooth, connected, affine curve defined over k with function field K and by π1(C) its algebraic fundamental group. In the original paper we addressed the question of the structure of the normal closed subgroups of π1(C). More precisely, in Theorem 1.1, we gave a sufficient condition for such a normal closed subgroup N to be isomorphic to a subgroup of a free profinite group of countable rank. There we claimed to prove this under the hypothesis that the group M = π1(C)/N had an infinitely generated Sylow p-subgroup. However, as remarked in the paragraph after Corollary 1.2 of [1], this hypothesis is not enough. We would like to make this remark more explicit by stating (as noticed by those authors) the additional hypothesis needed to make Theorem 1.1 work. For each integer g ≥ 0 denote by Pg(C) the intersection of algebraic fundamental groups

Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups

Given a free metabelian group S of finite rank r, r ≥ 2, we prove that a system of elements g1, ..., gn ∈ S for n = 1 or n = r preserves measure on the variety of all metabelian groups if and only if the system is primitive. Similar results hold for a free profinite group \(\hat S\) and the variety of finite metabelian groups for each n, 1 ≤ n ≤ r. Some corollaries to these theorems are derived.

Toward a theorem of douady

A theorem of Douady says that the absolute Galois group of a rational function field F(x) in one variable over an algebraically closed field F of characteristic 0 is a free profinite group. A new method is proposed to extend Douady’s theorem from the case of the complex number field F = ℂ to the case of an arbitrary field.

Profinite groups associated to sofic shifts are free

We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the minimal ideal of the free profinite semigroup and the J-class associated to an irreducible sofic shift.

Maximal subgroups of the minimal ideal of a free profinite monoid are free

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 H is variety of finite groups closed under extension and containing ℤ/pℤ for infinitely may primes p, the corresponding result holds for free pro-\( \bar H \) monoids.

Rational codes and free profinite monoids

It is well known that clopen subgroups of finitely generated free profinite groups are again finitely generated free profinite groups. Clopen submonoids of free profinite monoids need not be finitely generated nor free. Margolis, Sapir and Weil proved that the closed submonoid generated by a finite code (which is, in fact, clopen) is a free profinite monoid generated by that code. In this note we show that a clopen submonoid is free profinite if and only if it is the closure of a rational free submonoid. In this case its unique closed basis is clopen, and is, in fact, the closure of the corresponding rational code. More generally, our results apply to free pro- H ¯ monoids for H an extension-closed pseudovariety of groups.

Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (respectively, characteristic) subgroup of π 1(C). Under the hypothesis that the quotient π 1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (respectively, characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of N is a free profinite group of countable cardinality.

Note sur la détermination algébrique du groupe fondamental pro-résoluble d'une courbe affine

Let X be a smooth projective algebraic curve of genus g minus r ⩾ 1 points defined over an algebraically closed field k of characteristic p ⩾ 0 . The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods: it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2 g + r − 1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.