Open subgroups of free topological groups

TL;DR

This paper extends classical subgroup theorems to free topological groups using semicovering space theory, showing that open subgroups inherit freeness under certain conditions.

Contribution

It introduces a topological analogue of the Nielsen-Schreier theorem for free topological groups via semicovering spaces.

Findings

Open subgroups of free Graev topological groups are free Graev topological groups.

Open subgroups of free Markov topological groups are free if and only if disconnected.

The work generalizes classical subgroup theorems to topological group settings.

Abstract

The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.