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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-1-2,
author = {Jeremy Brazas},
title = {Open subgroups of free topological groups},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {17-40},
zbl = {1303.22001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-1-2}
}
Jeremy Brazas. Open subgroups of free topological groups. Fundamenta Mathematicae, Tome 227 (2014) pp. 17-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm226-1-2/