Narrow Results

We investigate the ability of a free pro-$\mathcal{𝒞}$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\mathcal{𝒞}$, the profinite completion of any order, of a free pro-$\mathcal{𝒞}$ group of infinite rank, is a free pro-$\mathcal{𝒞}$ group as well, of the corresponding rank.

We prove some new cases of the unramified Fontaine–Mazur conjecture for $p$-adic Galois representations.

A group Γ is called boundedly generated (BG) if it is the set-theoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations.

We use this to reprove and generalize Rapinchuk's theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension over any field. We also give a structure theorem for the profinite completion G of such a group Γ.

On the other hand, we exhibit boundedly generated profinite FAb groups which do not satisfy this structure theorem.

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[\dots [[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ 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 $G$, we prove that if, for some $n$, $|E_n(g)|\le m$ for all $g\in G$, then the order of the nilpotent residual ${\gamma }_{\infty }(G)$ is bounded in terms of $m$.

We classify the finitely generated prosupersolvable groups that satisfy Schreier’s formula for the number of generators of open subgroups.

Let $\mathrm{\Gamma }$ be a group and ${({\mathrm{\Gamma }}_n)}_{n=1}^{\infty }$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $\mathrm{\Gamma }/{\mathrm{\Gamma }}_n$, under some natural hypotheses on the behavior of power and commutator words in $\mathrm{\Gamma }$. The bounds we obtain do not depend on a choice of generating set. Moreover, under reasonable conditions our method provides a fast algorithm for navigating directed Cayley graphs. The proof is closely analogous to the Solovay–Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. We apply our procedure to give new directed diameter bounds on finite quotients of a large class of regular branch groups, and of ${\mathrm{\text{SL}}}_2({𝔽}_q[[t]])$ (for $q$ even).

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)={\sum }_{n\in \mathbb{N}}{a}_n(G)/n^s$ is associated, where $a_n(G)={\sum }_{|G:H|=n}\mu (H,G)$ and $\mu (H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse ${(P_G(s))}^{-1}$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of ${(P_G(s))}^{-1}$ is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

Our main goal is to study the freeness of Schützenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with corresponding non-free Schützenberger group. This constitutes a counterexample to a result of Almeida dating back to 2005. We also give some early results concerning relative freeness of Schützenberger groups, a question which remains largely unexplored.

The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer $r$ we prove that if $G$ is a profinite group in which the centralizer of every nontrivial element has rank at most $r$, then $G$ is either a pro-$p$ group or a group of finite rank. Further, if $G$ is not virtually a pro-$p$ group, then $G$ is virtually of rank at most $r+1$.

We prove the Cartan–Eilenberg stable elements theorem and construct a Lyndon–Hochschild–Serre type spectral sequence for pro-fusion systems. As an application, we determine the continuous mod-$p$ cohomology ring of ${\mathrm{\text{GL}}}_2({\mathbb{Z}}_p)$ for any odd prime $p$.

An Engel sink of an element $g$ of a group $G$ is a set $\mathcal{ℰ}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to $\mathcal{ℰ}(g)$. (Thus, $g$ is an Engel element precisely when we can choose $\mathcal{ℰ}(g)=\{1\}$.) It is proved that if every element of a compact (Hausdorff) group $G$ has a countable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. This settles a question suggested by J. S. Wilson.