Free non-archimedean topological groups

TL;DR

This paper explores free non-archimedean topological groups over uniform spaces, revealing their metrizability and continuous actions, and applying these findings to epimorphism density and universal group constructions.

Contribution

It provides new descriptions of free non-archimedean groups, showing their metrizability and continuous actions, and applies these to epimorphism density and universal group examples.

Findings

Free non-archimedean groups are metrizable under certain conditions.

Topological group actions on free abelian NA groups often remain continuous.

Any epimorphism in the NA category must be dense.

Abstract

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA groups frequently remain continuous. One of the main applications is: any epimorphism in the category NA must be dense. Moreover, the same methods improve the following result of T.H. Fay : the inclusion of a proper open subgroup H into G is not an epimorphism in the category of all Hausdorff topological groups. A key tool in the proofs is Pestov's test of epimorphisms. Our results provide a convenient way to produce surjectively universal NA abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao and Gao-Xuan as well as classical results about profinite groups which go back to Iwasawa and Gildenhuys-Lim.