Free product of Demushkin groups as absolute Galois group

TL;DR

This paper characterizes when free products of countably many Demushkin groups, each realizable as an absolute Galois group, form an absolute Galois group themselves, based on the growth of their invariants.

Contribution

It provides a precise criterion involving the growth of invariants for free products of Demushkin groups to be absolute Galois groups.

Findings

The free product of Demushkin groups is an absolute Galois group if and only if their invariants tend to infinity.

Establishes a link between group invariants and Galois realizability.

Extends understanding of the structure of absolute Galois groups in profinite settings.

Abstract

We prove that a free profinite (pro-$p$) product over a set converging to 1 of countably many Demushkin groups of rank $\aleph_0$, $G_i$, that can be realized as absolute Galois groups, is isomorphic to an absolute Galois group if and only if $\log_pq(G_i)\to \infty$.