Locally Minimal Topological Groups 2

TL;DR

This paper extends the concept of local minimality in topological groups, characterizes certain classes of compact abelian groups with dense locally minimal subgroups, and introduces the new class of almost minimal groups with various properties.

Contribution

It generalizes minimality criteria to local minimality, characterizes classes of compact abelian groups with dense locally minimal subgroups, and introduces the concept of almost minimal groups.

Findings

Characterization of compact abelian groups with dense locally minimal subgroups.

Introduction of almost minimal groups and their properties.

Examples of countable precompact metrizable locally minimal groups that are not almost minimal.

Abstract

We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group $G$ {\it almost minimal} if it has a closed, minimal normal subgroup $N$ such that the quotient group $G/N$ is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.