Subgroups of free proalgebraic groups and Matzat's conjecture for function fields

TL;DR

This paper proves that finite index subgroups of free proalgebraic groups are also free, and applies this to show the absolute differential Galois group of a one-variable function field is free, advancing differential Galois theory.

Contribution

It establishes that finite index subgroups of free proalgebraic groups are free, with applications to differential Galois groups of function fields.

Findings

Finite index subgroups of free proalgebraic groups are free.

The absolute differential Galois group of a one-variable function field is free.

Supports Matzat's conjecture in the context of function fields.

Abstract

We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. Our main motivation for this result is an application in differential Galois theory: The absolute differential Galois group of a one-variable function field is a free proalgebraic group.