Profinite Groups Research Articles

Commutators and commutator subgroups in profinite groups

Let G be a profinite group. We prove that the commutator subgroup G′ is finite-by-procyclic if and only if the set of all commutators of G is contained in a union of countably many procyclic subgroups.

Profinite groups with NIP theory andp-adic analytic groups

We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.

PRESENTATIONS: FROM KAC-MOODY GROUPS TO PROFINITE AND BACK

We go back and forth between, on the one hand, presentations of arithmetic and Kac-Moody groups and, on the other hand, presentations of profinite groups, deducing along the way new results on both.

An axiomatizable profinite group with infinitely many open subgroups of index 2

Abstract 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 2 ⁢ p {2p} , is necessarily isomorphic to this direct product.

Hausdorff dimension in R-analytic profinite groups

Abstract We study the Hausdorff dimension of R-analytic subgroups in an infinite R-analytic profinite group, where R is a pro-p ring whose associated graded ring is an integral domain. In particular, we prove that the set of such Hausdorff dimensions is a finite subset of the rational numbers.

Profinite and discrete G–spectra and iterated homotopy fixed points

For a profinite group $G$, let $(\text{-})^{hG}$, $(\text{-})^{h_dG}$, and $(\text{-})^{h'G}$ denote continuous homotopy fixed points for profinite $G$-spectra, discrete $G$-spectra, and continuous $G$-spectra (coming from towers of discrete $G$-spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K \vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ exist and are $X^{hG}$. For the Lubin-Tate spectrum $E_n$ and $G <_c G_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G/K$-spectrum with $(E_n^{hK})^{hG/K} \simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{h'K})^{h'G/K} \simeq E_n^{h'G}$ or the Devinatz-Hopkins proof (which requires $|G/K| < \infty$) of $(E_n^{dhK})^{h_dG/K} \simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G/K$-homotopy fixed point spectral sequence for $\pi_\ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; \pi_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $\pi_\ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; \pi_t(E_n^{dhK}))$, and the descent spectral sequence for $\pi_\ast((E_n^{h'K})^{h'G/K})$.

Introduction to Brauer theory for profinite groups

A brief introduction to Brauer Theory, emphasizing functorial aspects that allow for a natural extension to profinite groups. Based on joint work with Peter Symonds.

Almost Engel finite and profinite groups

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] 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 [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

The inverse deformation problem

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$, we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$.

Determining Fuchsian groups by their finite quotients

Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.

On strongly just infinite profinite branch groups

Abstract For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger–Mozes universal simple groups enjoy several automatic continuity properties.

On the free profinite products of profinite groups with commuting subgroups

In this paper we introduce the construction of free profinite products of profinite groups with commuting subgroups‎. ‎We study a particular case‎: ‎the proper free profinite products of profinite groups with commuting subgroups‎. ‎We prove some conditions for a free profinite product of profinite groups with commuting subgroups to be proper‎. ‎We derive some consequences‎. ‎We also compute profinite completions of free products of (abstract) groups with commuting subgroups‎.

Bieri–Eckmann criteria for profinite groups

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FPn 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 FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FPn but not of type FPn+1 over ZĈ for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FPn fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.

A geometric interpretation of the Schützenberger group of a minimal subshift

The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berth\'e et. al.

Prosolvability criteria and properties of the prosolvable radical via Sylow sequences

Abstract We extend a finite group solvability criterion of J. G. Thompson, based on his classification of finite minimal simple groups, to a prosolvability criterion. Moreover, we generalize to the profinite setting subsequent developments of Thompson's criterion by G. Kaplan and the second author, which recast it in terms of properties of sequences of Sylow subgroups and their products. This generalization also encompasses a possible characterization of the prosolvable radical whose scope of validity is still open even for finite groups. We prove that if this characterization is valid for finite groups, then it carries through to profinite groups.

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

A second countable virtually free pro-\(p\) group all of whose torsion elements have finite centralizer is the free pro-\(p\) product of finite p-groups and a free pro-\(p\) factor. The proof explores a connection between p-adic representations of finite p-groups and virtually free pro-p groups. In order to utilize this connection, we first prove a version of a remarkable theorem of A. Weiss for infinitely generated profinite modules that allows us to detect freeness of profinite modules. The proof now proceeds using techniques developed in the combinatorial theory of profinite groups. Using an HNN-extension, we embed our group into a semidirect product \(F\rtimes K\) of a free pro-p group F and a finite p-group K that preserves the conditions on centralizers and such that every torsion element is conjugate to an element of K. We then show that the \(\mathbb {Z}_pK\)-module F / [F, F] is free using the detection theorem mentioned above. This allows us to deduce the result for \(F\rtimes K\), and hence for our original group, using the pro-p version of the Kurosh subgroup theorem.

On profinite groups of type FP∞

Suppose R is a profinite ring. We construct a large class of profinite groups L′HRˆF, including all soluble profinite groups and profinite groups of finite cohomological dimension over R. We show that, if G∈L′HRˆF is of type FP∞ over R, then there is some n such that HRn(G,R〚G〛)≠0, and deduce that torsion-free soluble pro-p groups of type FP∞ over Zp have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.

On conciseness of words in profinite groups

Let w be a group word. It is conjectured that if w has only countably many values in a profinite group G, then the verbal subgroup w(G) is finite. In the present paper we confirm the conjecture in the cases where w is a multilinear commutator word, or the word x2, or the word [x2,y].

The inverse problem for universal deformation rings and the special linear group

We present a solution to the inverse problem for universal deformation rings of group representations. Namely, we show that every complete noetherian local commutative ring R R with a finite residue field k k can be realized as the universal deformation ring of a continuous linear representation of a profinite group. More specifically, R R is the universal deformation ring of the natural representation of S L n ( R ) \mathrm {SL}_n(R) in S L n ( k ) \mathrm {SL}_n(k) , provided that n ≥ 4 n\geq 4 . We also check for which R R an analogous result is true in case n = 2 n=2 and n = 3 n=3 .

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.