Profinite Groups Research Articles

Profinite groups with restricted centralizers of $$\pi $$-elements

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 \(\pi \), we take interest in profinite groups with restricted centralizers of \(\pi \)-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-\(\pi \) subgroup and Q is a pro-\(\pi '\) subgroup. This significantly strengthens a result from our earlier paper.

Holomorphic Reflexivity for Locally Finite and Profinite Groups: The Abelian and General Cases

Предложенная Акбаровым теория голоморфной рефлексивности для топологических алгебр Хопфа получила развитие в двух направлениях - через усложнение определений при расширении области применимости и через их упрощение при сужении. В рамках второго подхода мы устанавливаем голоморфную рефлексивность для топологических алгебр Хопфа, ассоциированных с локально конечными счeтными группами и проконечными группами со второй аксиомой счетности. В абелевом случае рефлексивность описана в терминах, близким к классическим. Библиография: 23 названия.

Expansive dynamics on profinite groups

A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups.

On profinite groups with automorphisms whose fixed points have countable Engel sinks

An Engel sink of an element g of a group G is a set $${\cal E}(g)$$ such that for every x ∈ G all sufficiently long commutators [⋯[[x, g], g],…, g] belong to $${\cal E}(g)$$ . (Thus, g is an Engel element precisely when we can choose $${\cal E}(g) = \{ 1\} $$ .) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q2 for a prime q such that for each a ∈ A {1} every element of the centralizer CG(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underline{\pi}_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite \'etale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies $\operatorname{fced}\underline{\pi}_{1}(X)=\dim X=1$ for $X$ as above. We prove that, if $A$ is an abelian variety over a field finitely generated over $\mathbb{Q}$, then $\operatorname{fced}\underline{\pi}_{1}(A)=\operatorname{fced} TA=\dim A$.

Cantor dynamics of renormalizable groups

A group \Gamma is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding \varphi \colon \Gamma \to \Gamma whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for \Gamma .” In this work, we associate a dynamical system to a renormalization \varphi of \Gamma . The discriminant invariant {\mathcal D}_{\varphi} of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If {\mathcal D}_{\varphi} is a finite group for some renormalization, we show that \Gamma/C_{\varphi} is virtually nilpotent, where C_{\varphi} is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.

Block theory and Brauer's first main theorem for profinite groups

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra k[[G]] as a product ∏i∈IBi of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a pro-p subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a version of Brauer's first main theorem: a correspondence between the blocks of G with defect group D and the blocks of the normalizer NG(D) with defect group D.

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems.

Cohomology of profinite groups of bounded rank

We generalise to profinite groups some of our previous results on the cohomology of pro- p groups of bounded sectional p-rank.

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.

Finite axiomatizability for profinite groups

A group is finitely axiomatizable (FA) in a class C if it can be determined up to isomorphism within C by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups, or in the class of pro- p groups for some prime p. Both algebraic and model-theoretic methods are developed for the purpose. Reasons why certain groups cannot be FA are also discussed.

THE PROFINITE COMPLETION OF A PROFINITE PROJECTIVE GROUP

AbstractWe prove that the profinite completion of a profinite projective group is projective.

Coarse groups, and the isomorphism problem for oligomorphic groups

Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup G of [Formula: see text] is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of G, and a ternary relation [Formula: see text]. This structure derived from G was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3 ]. If G has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of G from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.

Profinite groups in which centralizers are virtually procyclic

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 (Theorem 1.1). The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic (Theorem 1.2); moreover, if G is not pro-p, then G has finite rank.

Geometric Structures in Group Theory

The conference focused on the use of geometric methods to study infinite groups and the interplay of group theory with other areas. One of the central techniques in geometric group theory is to study infinite discrete groups by their actions on nice, suitable spaces. These spaces often carry an interesting large-scale geometry, such as non-positive curvature or hyperbolicity in the sense of Gromov, or are equipped with rich geometric or combinatorial structure. From these actions one can investigate structural properties of the groups. This connection has become very prominent during the last years. In this context non-discrete topological groups, such as profinite groups or locally compact groups appear quite naturally. Likewise, analytic methods and operator theory play an increasing role in the area.

The Karoubi envelope of the mirage of a subshift

We study a correspondence associating to each subshift of a subcategory of the Karoubi envelope of the free profinite semigroup generated by A. The objects of this category are the idempotents in the mirage of that is, in the set of pseudowords whose finite factors are blocks of The natural equivalence class of the category is shown to be invariant under flow equivalence. As a corollary of our proof, we deduce the flow invariance of the profinite group that Almeida associated to each irreducible subshift. We also show, in a functorial manner, that the isomorphism class of the category is invariant under conjugacy. Finally, we see that the zeta function of is naturally encoded in the category. These results hold, with obvious translations, for relatively free profinite semigroups over many pseudovarieties, including all of the form with a pseudovariety of groups.

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).

Lifting low-dimensional local systems

Let k be a field of characteristic \(p>0\). Denote by \(\mathbf {W}_r(k)\) the ring of truntacted Witt vectors of length \(r \ge 2\), built out of k. In this text, we consider the following question, depending on a given profinite group G. Q(G): Does every (continuous) representation \(G\longrightarrow \mathrm {GL}_d(k)\) lift to a representation \(G\longrightarrow \mathrm {GL}_d(\mathbf {W}_r(k))\)? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in De Clercq and Florence (https://arxiv.org/abs/2009.11130, 2018) under the name “smooth profinite groups”. Using Grothendieck-Hilbert’ theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over \(\mathbb {Z}[\frac{1}{p}]\), smooth curves over algebraically closed fields, and affine schemes over \(\mathbb {F}_p\). In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to Q(G), for a cyclotomic profinite group G: the answer is positive, when \(d=2\) and \(r=2\). When \(d=2\) and \(r=\infty \), we show that any 2-dimensional representation of G stably lifts to a representation over \(\mathbf {W}(k)\): see Theorem 6.1. When \(p=2\) and \(k=\mathbb {F}_2\), we prove the same results, up to dimension \(d=4\). We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).

The weight of nonstrongly complete profinite groups

Abstract We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set {{mathscr {R}}}(g) such that for every xin 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 varphi of prime order such that every fixed point of varphi 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 xin G all sufficiently long commutators [...[[x,g],g],dots ,g] belong to {{mathscr {E}}}(g). (Thus, g is a left Engel element precisely when we can choose {mathscr {E}}(g)={ 1}.) We prove that if a profinite group G admits a coprime automorphism varphi of prime order such that every fixed point of varphi has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.