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For a prime number $p$ and a free pro-$p$ group $G$ on a totally ordered basis $X$, we consider closed normal subgroups $G^\Phi$ of $G$ which are generated by $p$-powers of iterated commutators associated with Lyndon words in the alphabet…

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. We take…

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and…

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and…

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\Gamma$ be a finite simplicial graph and $G = G_{\Gamma}$ be the corresponding pro-$\mathcal C$ RAAG. We show that if $N$ is a…

The Pr\"ufer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits…

We study the \'{e}tale fundamental groups of singular reduced connected curves defined over an algebraically closed field of arbitrary prime characteristic. It is shown that when the curve is projective, the \'{e}tale fundamental group is a…

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring…

Zassenhaus filtrations of profinite groups are an important tool to study profinite groups.In this paper, we describe Zassenhaus filtrations of profinite groups as intersections of kernels of certain representations. In this way we…

Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p \geq 0$ with algebraic closure $\bar{K}$, and let $\alpha_{1}, ..., \alpha_{d} \in \mathbb{P}_{K}^{1}(K)$. In this paper, we give explicit generators and…

We investigate some properties of the $p$-elements of a profinite group $G$. We prove that if $p$ is odd and the probability that a randomly chosen element of $G$ is a $p$-element is positive, then $G$ contains an open prosolvable subgroup.…

We give a criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which p-kernels occur in an isogeny class of p-divisible groups over an algebraically closed field of positive…

Let $\pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of…

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be…

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a continuous representation $\theta\colon G\to\mathrm{GL}_1(\mathbb{Z}_p)$ such that every open subgroup $H$ of $G$, together with the restriction…

Let F be a number field and p be a prime. In the Successive Approximation Theorem, we prove that, for each positive integer n, finitely many candidates for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower…

We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a…

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…

For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $\chi(S)=2-2g-n<0$, let $\mathrm{P}\Gamma(S)$ be the pure mapping class group of $S$ and…

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure…