Related papers: On the Bogomolov-Positselski Conjecture

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the…

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod…

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either…

It is well known that on arbitrary metric measure spaces, the notion of minimal $p$-weak upper gradient may depend on $p$. In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded…

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

Let $p$ be an odd prime and $L/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to…

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to…

Let G be a nilpotent p-valuable (compact p-adic Lie) group. There is an ongoing investigation into the prime ideals of its completed group algebra (Iwasawa algebra), and there remains an open conjecture that they can all be proved to have a…

It is known that the Kurosh Subgroup Theorem does not hold for pro-$p$ groups of big cardinality. However, a subgroup $G$ of a free pro-$p$ product is projective relative to the Kurosh family of subgroups. In this paper we prove the…

We develop an abstract framework for studying the strong form of Malle's conjecture for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle's conjecture for any nilpotent group…

Let $p$ be a prime. We resolve a question posed by Min\'a\v{c}-Rogelstad-T\^an. We relate the Zassenhaus and the lower central series of pro-$p$ groups under a torsion-freeness condition. We also study graph products of (pro-$p$) groups…

We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A…

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of…

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm…

The purpose of this note is to show that, in contrast to the ${\Bbb F}_p$-case (proven in [7]), the $p$-primary and $p$-adic cases of the Isotropy Conjecture, claiming that the isotropic Chow groups with ${\Bbb Z}/p^r$, $r>1$, respectively,…

We prove a criterion for the mildness of a finitely presented pro-$p$ group $G$. It implies as a special case a cohomological mildness criterion via Massey products, generalizing results due to Schmidt and G\"artner. It subsumes Labute's…

For any odd prime $p$ and any imaginary quadratic field $K$, the $p$-tower group $G_K$ associated to $K$ is the Galois group over $K$ of the maximal unramified pro-$p$-extension of $K$. This group comes with an action of a finite group…

This is a survey of two papers joint with A. Borisov and a paper joint with I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews 2009", Bath (August 2009). We prove that almost all 1-related groups with at least 3…

Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…