Galois-theoretic features for 1-smooth pro-$p$ groups

Abstract

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 $\theta :G\to {\mathrm{GL}}_1({\mathbb{Z}}_p)$ such that every open subgroup $H$ of $G$, together with the restriction $\theta {|}_H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-$p$ group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-$p$ Galois groups of fields, and that if a 1-smooth pro-$p$ group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-$p$ Galois groups of fields. Finally we ask whether 1-smooth pro-$p$ groups satisfy a "Tits' alternative".

To appear on "Canadian Math. Bull"

References in corpus (3)

• Bloch-Kato pro-p groups and locally powerful groups
• Lifting Theorems and Smooth Profinite Groups
• Oriented pro-$\ell$ groups with the Bogomolov-Positselski property

Cited by in corpus (4)

• Groups of p-absolute Galois type that are not absolute Galois groups
• Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups
• Chasing maximal pro-p Galois groups via 1-cyclotomicity
• Frattini-injectivity and Maximal pro-$p$ Galois groups