Profinite groups with a cyclotomic $p$-orientation

Abstract

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 cyclotomic $p$-orientation ${\theta }_{K,p}:G_K\to {\mathbb{Z}}_p^{\times }$ which is even Bloch-Kato. The same is true for its maximal pro-$p$ quotient $G_K(p)$ provided the field $K$ contains a primitive $p^{th}$-root of unity. The class of cyclotomically $p$-oriented profinite groups (resp. pro-$p$ groups) which are Bloch-Kato is closed with respect to inverse limits, free product and certain fibre products. For profinite groups with a cyclotomic $p$-orientation the classical Artin-Schreier theorem holds. Moreover, Bloch-Kato pro-$p$ groups with a cyclotomic orientation satisfy a strong form of Tits' alternative, and the elementary type conjecture formulated by I. Efrat can be restated that the only finitely generated indecomposable torsion free Bloch-Kato pro-$p$ groups with a cyclotomic orientation should be Poincaré duality pro-$p$ groups of dimension less or equal to $2$.

To appear on "Doc. Math"

References in corpus (6)

• Norm varieties and algebraic cobordism
• The Kummerian Property and Maximal Pro-$p$ Galois Groups
• Bloch-Kato pro-p groups and locally powerful groups
• Lifting Theorems and Smooth Profinite Groups
• The Bloch-Kato Conjecture and Galois Theory
• Right-angled Artin pro-$p$ groups

Cited by in corpus (8)

• The Kummerian Property and Maximal Pro-$p$ Galois Groups
• Koszul algebras and quadratic duals in Galois cohomology
• On pro-$p$ groups with quadratic cohomology
• Galois-theoretic features for 1-smooth pro-$p$ groups
• Mild pro-p groups and the Koszulity conjectures
• Right-angled Artin pro-$p$ groups
• One relator maximal pro-p Galois groups and the Koszulity conjectures
• Frattini-injectivity and Maximal pro-$p$ Galois groups