The kernel generating condition and absolute Galois groups

TL;DR

This paper provides a cohomological framework to characterize certain subgroups of profinite groups, especially absolute Galois groups, using the kernel generating condition and intersections of open normal subgroups related to finite groups.

Contribution

It introduces a cohomological characterization of the kernel generating condition and applies it to describe specific filtrations of absolute Galois groups.

Findings

Cohomological description of the kernel generating condition.

Application to the $p$-Zassenhaus and lower $p$-central filtrations.

Recovery of known results for absolute Galois groups.

Abstract

For a list $\cal{L}$ of finite groups and for a profinite group $G$, we consider the intersection $T(G)$ of all open normal subgroups $N$ of $G$ with $G/N$ in $\cal{L}$. We give a cohomological characterization of the epimorphisms $\pi\colon S\to G$ of profinite groups (satisfying some additional requirements) such that $\pi[T(S)]=T(G)$. For $p$ prime, this is used to describe cohomologically the profinite groups $G$ whose $n$th term $G_{(n,p)}$ (resp., $G^{(n,p)}$) in the $p$-Zassenhaus filtration (resp., lower $p$-central filtration) is an intersection of this form. When $G=G_F$ is the absolute Galois group of a field $F$ containing a root of unity of order $p$, we recover as special cases results by Minac, Spira and the author, describing $G_{(3,p)}$ and $G^{(3,p)}$ as $T(G)$ for appropriate lists $\cal{L}$.