The Structure of Sequentially Complete Locally Minimal Groups

TL;DR

This paper investigates the structure of locally minimal Abelian groups, establishing conditions under which their connected components match those of their ambient locally compact groups, and explores special classes called critical locally minimal groups.

Contribution

It generalizes previous results to dense locally minimal subgroups of LCA groups, characterizes their connected components, and introduces the class of critical locally minimal groups with detailed structural properties.

Findings

Connected component of dense locally minimal subgroups matches that of the ambient LCA group.

When the weight is not Ulam measurable, the connected components are equal.

Complete description of connected components in finite-dimensional groups within a specific class of compact groups.

Abstract

Generalizing results from \cite{DTk,DU} we study the fine structure of locally minimal (locally) precompact Abelian groups (these are the locally essential subgroups $G$ of LCA groups $L$, i.e., such that $G$ non-trivially meets all ``small" closed subgroup of $L$). More precisely we prove that if $G$ is a dense locally minimal and sequentially closed subgroup of a LCA group $L$, then the connected component $c(G)$ of $G$ has the same weight as $c(L)$. Moreover, when $w(c(G))$ is not Ulam measurable, then $c(G) = c(L)$. We provide an extended discussion illustrating how this result fails in various ways in the non-abelian case (even for nilpotent groups of class 2). Motivated by the above result, we study further those locally minimal precompact Abelian groups $G$, termed {\em critical locally minimal},such that $c(G) =c(K)$ (where $K$ is the compact completion of $G$) and $G/c(G)$ is not locally minimal. Such a group cannot be compact, neither connected, nor totally disconnected. We provide a proper class of critical locally minimal groups with additional compactness-like properties and we study the class $\CCC$ of compact Abelian groups with a dense critical locally minimal subgroup. In particular, we completely describe the connected components of the finite-dimensional groups belonging to $\CCC$.