Free Profinite Group Research Articles

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let kυ be the completion of k with respect to υ a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kυ-rank 1. Let G = G(kυ). Let Г be an arithmetic lattice in G and let C = C(Г) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Г by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example , where is the ring of S-integers in k, with S = {υ}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Г on the Bruhat-Tits tree associated with G.

The absolute Galois groups of finite extensions of $${\mathbb{R}}(t)$$

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ≅ Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card (R) if L is not formally real.

Profinite groups associated with weakly primitive substitutions

A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword that is a factor of all its infinite factors, i.e., one that lies in a \(\mathcal{J}\)-class with only finite words strictly \(\mathcal{J}\)-above it. Such a \(\mathcal{J}\)-class is regular, and therefore it has an associated profinite group, namely, any of its maximal subgroups. One way to produce such \(\mathcal{J}\)-classes is to iterate finite weakly primitive substitutions. This paper is a contribution to the computation of the profinite group associated with the \(\mathcal{J}\)-class that is generated by the infinite iteration of a finite weakly primitive substitution. The main result implies that the group is a free profinite group provided the substitution induced on the free group on the letters that appear in the images of all of its sufficiently long iterates is invertible.

Diamond theorem for a finitely generated free profinite group

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 of Fried and Jarden (in Field Arithmetic, 2nd edn, Springer, Berlin Heidelberg, New York, 2005).

Profinite surface groups and the congruence kernel of arithmetic lattices in SL2 (R)

LetX be a proper, nonsingular, connected algebraic curve of genusg over the fieldC of complex numbers. The algebraic fundamental group Γ = π1 (X) in the sense of [SGA-1] (1971) is the profinite completion of the fundamental group π1top (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, characteristic, accessible) subgroup of Γ is isomorphic to a normal (respectively, characteristic, accessible) subgroup of a free profinite group. We use this description to give a complete solution of the congruence subgroup problem for arithmetic lattices in SL2 (R).

On Certain Closed Normal Subgroups of Free Profinite Groups of Countably Infinite Rank#

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R Δ(F) is solvable.

A Fundamental Group for Dynamical Systems III: Simple Compactifications

This paper continues exploration of a recent homotopy construction for group-sets. It characterizes the fundamental group of the standard one point and two point compactifications of . Each is a free profinite group on two generators. However, the canonical morphism between them does not induce an isomorphism on homotopy groups.

The geometry of profinite graphs with applications to free groups and finite monoids

We initiate the study of the class of profinite graphsΓ\Gammadefined by the following geometric property: for any two verticesvvandwwofΓ\Gamma, there is a (unique) smallest connected profinite subgraph ofΓ\Gammacontaining them; such graphs are calledtree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to bedendralif it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groupsH\mathbf {H}to bearboreousif all finitely generated free pro-H\mathbf {H}groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarietiesH\mathbf {H}, a pro-H\mathbf {H}analog of the Ribes and Zalesskiĭ product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutionsH\mathbf {H}to the much studied pseudovariety equation\mathbf {J}\ast \mathbf {H}= \mathbf {J} \text {\textcircled {m}} \mathbf {H}.

A note on the continuous extensions of injective morphisms between free groups to relatively free profinite groups

Let V be a pseudovariety of finite groups such that free groups are residually V, and let ϕ: F(A) → F(B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension ˆϕ of ϕ between the pro-V completions of F(A) and F(B) is also injective. In particular, if V is extension-closed, this is the case if and only if ϕ(F(A)) and its pro-V closure in F(B) have the same rank. We examine a number of situations where the injectivity of ˆϕ can be asserted, or at least decided, and we draw a few corollaries.

Pro-finite Presentations

The goal of this paper is to start to present a systematic presentation theory for pro-finite groups. Ž . Let G be a fixed finitely-generated pro-finite group and P a free presentation of it; i.e., P 1 R F G 1 Ž . is an exact sequence when F is a finitely generated free pro-finite group. Ž . Ž . We write d F for the minimal number of generators of F and d R for F the minimal number of generators of R as a normal subgroup of F. In G2 Gruenberg presented a systematic study of presentation theory Ž . of a discrete group G mainly for a finite group G and presented three basic problems: Ž . QUESTION 1. Given P 1 R F G 1 for i 1, 2, when F is i i Ž . Ž . a free group, is then d R d R ? F 1 F 2

Random normal subgroups of free profinite groups

Random normal subgroups of free profinite groups

Free subgroups of free profinite groups

Free subgroups of free profinite groups

Finitely generated subgroups of free profinite groups and of some Galois groups

It is proved that, for a (closed) subgroupH of a free profinite or free prosolvable groupF of rankF>1 such thatH contains a nontrivial composition subgroupN ofF, we have rankF<∞ and [F:H]<∞.

Large normal extension of Hilbertian fields

The goal of this note is to consider a certain natural family of closed normal subgroups of G(Q) and to prove that each group in this family is free. More generally, consider a countable separably Hilbertian field K . Denote the absolute Galois group of K by G(K ). Then, for almost all σ ∈ G(K )e the field Ks (σ) is PAC and e-free [FJ2, Thms. 16.13 and 16.18]. Here Ks is the separable closure of K and Ks (σ) is the fixed field of σ in Ks . Being PAC means that every nonvoid absolutely irreducible variety defined over Ks (σ) has a Ks (σ)-rational point. We say that Ks (σ) is e-free if G(Ks (σ)) (i.e., the closed subgroup 〈σ1, . . . , σe〉 of G(K ) generated by σ1, . . . , σe) is free on e generators. Denote the largest Galois extension of K which is contained in Ks (σ) by Ks [σ]. It is the intersection of all K -conjugates of Ks (σ) and also the fixed field of the smallest closed normal subgroup of G(K ) which contains σ1, . . . , σe . If char(K ) = 0, then, for almost all σ ∈ G(K )e the field Ks [σ] is PAC [FJ2, Thm. 16.47]. Lemma 1.2 below generalizes this result to arbitrary characteristic. If we knew that Ks [σ] is separably Hilbertian, then a theorem of Fried-Volklein and Pop would imply that Ks [σ] is ω-free. That is, G(Ks [σ]) is isomorphic to the free profinite group Fw on countably many generators. Unfortunately, it is not clear how to prove the Hilbertianity of almost all Ks [σ] directly. So, we use instead a forerunner to the above mentioned theorem of Fried-Volklein-Pop and a recent theorem of Neumann [Neu] to prove directly that G(Ks [σ]) ∼= Fω for almost all σ ∈ G(K )e . A theorem of Roquette, then implies that Ks [σ] is also separably Hilbertian. Let K be the algebraic closure of K . Denote the maximal purely inseparable extension of Ks [σ] by K [σ]. Then, for almost all σ ∈ G(K )e , the field K [σ]

REDUCED FACTORIZATIONS IN FREE PROFINITE GROUPS AND JOIN DECOMPOSITIONS OF PSEUDOVARIETIES

In this paper we investigate certain notions from the theory of free profinite groups, which we then apply to the study of certain pseudovarieties of monoids. More precisely, the profinite Cayley graphs of the free pro-H groups are H-trees for certain, so-called arborescent, pseudovarieties of groups H. For such pseudovarieties, we introduce the notion of reduced factorizations in the free pro-H groups, and we prove that these factorizations enjoy a common refinement property. This property, in turn, is used to compute joins of certain pseudovarieties of monoids with pseudovarieties of groups within the pseudovariety DS of all finite monoids in which each regular [Formula: see text]-class is a subsemigroup. We then apply our results to the theory of formal languages.

The Embedding Problem Over a Hilbertian PAC-Field

We show that the absolute Galois group of a countable Hilbertian P(seudo)- A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., � ∞=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RG-Hilbertian property

The elementary theory of free pseudo p-adically closed fields of finite corank

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

Frobenius subgroups of free products of prosolvable groups

In this paper we establish the existence of profinite Frobenius subgroups in a free prosolvable productA ∐B of two finite groupsA andB. In this way the classification of solvable subgroups of free profinite groups is completed.

Subgroups of free pro-p-products

If F is a free profinite group, it is well known that the closed subgroups of F need not be free profinite; however, if p is a prime number, every closed subgroup of a free pro-p-group is free pio-p (cf. [2, 8, 7]). In this paper we show that there is an analogous contrast regarding the closed subgroups of free products in the category of profinite groups, and the closed subgroups of free products in the category of pro-p-groups, at least for (topologically) finitely generated subgroups.

Decidable regularly closed fields of algebraic numbers

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.