On Closed Subgroups of Free Products of Profinite Groups

Abstract

IntroductionIf G is a free product of a family {Ai}ieI of discrete groups then a subgroup H of G is the free product of a free group F and (Af n H), where a E ~(i), i E I, and ~(i) is a set of representatives of Ai \ G / H. This is the content of the Kurosh subgroup theorem (KST). Is a similar result true for closed subgroups of free (profinite) products of pro finite groups? (Say, with F projective instead of free.)An answer to this question requires an appropriate definition of a free product over an infinite family of groups. Such a definition has been proposed, by Gildenhuys and Ribes in [3], for groups indexed by compact topological spaces so that the factors are locally equal to each other, except for neighbourhoods of one distinguished point. In spite of the fact that the KST holds for open subgroups of such free products, this definition seems to be too restrictive: if H is a closed subgroup of the free product then the groups Af n H, with a E G, i E I, need not be 'locally equal' to each other (cf. Example 2.4).We propose a very natural generalization of the free product with finitely many factors: an inverse limit of such free products (over an inverse system with mappings that send respective factors again into factors of a free product). This, essentially, also includes the definition of [3].We do not know whether the analogue of the KST holds for open subgroups of these free products. Nevertheless, if we restrict ourselves to separable groups, we give a satisfactory account of the closed subgroups of the free products.1. The analogue of the KST does not hold, in general, for closed subgroups of free products (Example 5.5).2. We define for a pro finite group G the notion of projectivity relative to a given family I of its subgroups (Definition 4.2). We show:2a. if G is a free product of the groups in I, and H is a closed subgroup of G, then H is projective relative to {fa n HI f E I, a E G};2b. conversely, if H is separable and projective relative to ~ then H is a closed subgroup of a free product G of a family I of subgroups such that ~ = {fa n HI rEI, a E G}.3. Separable relative projective pro-p-groups are in fact free products (Corollary 9.6).Hence we can answer a question of Lubotzky [13, 2.10]: 4. The KST holds for separable closed subgroups of free pro-p-products.

Frobenius subgroups of free products of prosolvable groups

“…By Prop. 2.1 in [8] Next we extend the above result to a general free pro-qf-product G = I_I Ax of pro-C-groups Ax, indexed by a topological space X in X the sense of [4], [13], [14] or [3]. It is not difficult to prove that G is a projective limit of pro-~-groups G = lim Gi over a directed set I with ecanonical epimorphisms w;: G ~ G~ for ie/, and ~%: Gi -+ Gj for i ~> j, such that (1) each G~ is a free pro-Cg-product G~ = H G~k of a finite number of finite groups GjkEcg; (2) for every ieI and every xeX, ~i(Ax) <~ Gik for some kE {1,..., n3; (3) if i >~ j (in/), then ~;j maps every Gik into some Gjl.…”

“…Lemma 3.2. Let G = H Ax be a free pro-Cg-product of prof,-groups Ax in the sense of [4], [13], [3] or [14] Proof. Since ~0i(Ax) is finite for each i, if H is conjugate to a subgroup of some Ax,~0i(/-/) will also be finite for each ieL Conversely, assume that for every ie I, the group H,.…”

“…Define Xi = {xe G;I H x ~< Gik(0}; then Xi is obviously non-empty, and we assert that it is a compact set. For, let [4], [13], [14] or [3].…”

Prosolvable subgroups of free products of profinite groups

“…. , e. Moreover, by Haran's subgroup theorem [H,Thm. 5.1] each closed subgroup H of G is projective with respect to the family of the groups of the form G x i ∩ H, where x ranges over G and i = 1, .…”

“…By [H,Prop. 3.7 and Lemma 3.6] there is an etale space (E, X) and a free product ϕ: (E, X) → F such that X ⊆ X, ϕ(X) = X , and the restriction of ϕ to each Γ ∈ X is the identity map.…”