Related papers: Mild Pro-$p$ Groups and Ordered Monoids

We prove \cite[Conjecture~5.17]{Clausen} on the local light--profinite structure of smooth $p$-adic analytic Artin stacks. The argument proceeds in several reductions. First, by proving a generalization of van~Dantzig theorem for groupoids,…

Profinite groups with a cyclotomic $p$-orientation are introduced and studied. The special interest in this class of groups arises from the fact that any absolute Galois group $G_{K}$ of a field $K$ is indeed a profinite group with a…

We prove that for every prime $p$ algebraically clean graphs of groups are virtually residually $p$-finite and cohomologically $p$-complete. We also prove that they are cohomologically good. We apply this to certain $2$-dimensional Artin…

The homology groups introduced by A. Brumer can be used to establish a criterion ensuring that a profinite $\mathbb{F}_p[[G]]$-module of a pro-$p$ group $G$ has projective dimension $d<\infty$ (cf. Thm. A). This criterion yields a new…

The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups. Prime examples of such groups are the maximal Galois pro-$p$ groups of fields…

In [10] Benjamin Klopsch and Ilir Snopce posted the conjecture that for $p\geq 3$ and $G$ a torsion-free pro-$p$ group $d(G)=\dim (G)$ is a sufficient and necessary condition for the pro-$p$ group $G$ to be uniform. They pointed out that…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Using the mixed Lie algebras of Lazard, we extend the results of the first author on mild groups to the case p=2. In particular, we show that for any finite set S_0 of odd rational primes we can find a finite set S of odd rational primes…

We will prove the relative homotopy principle for smooth maps with singularities of a given {\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by…

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of…

We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the…

We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of…

Let $G$ be a finite nilpotent group and $n\in \{3,4, 5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where $G$ embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any…

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic…

We study 3-dimensional Poincar\'e duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index,…

Let $p$ be a prime. Following Snopce-Tanushevski, a pro-$p$ group $G$ is called Frattini-resistant if the function $H\mapsto\Phi(H)$, from the poset of all closed finitely-generated subgroups of $G$ into itself, is a poset embedding. We…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these…

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…

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new…