Permanence criteria for semi-free profinite groups

TL;DR

This paper introduces semi-free profinite groups, a new class that generalizes free groups, and demonstrates their permanence properties, leading to insights about absolute Galois groups over certain field extensions.

Contribution

It defines semi-free profinite groups, proves their permanence properties, and applies these results to Galois groups over field extensions of k((x,y)).

Findings

Semi-free groups generalize free groups.

Permanence properties extend to semi-free groups.

Many extensions of k((x,y)) have free Galois groups.

Abstract

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.