The Structure of Sequentially Complete Locally Minimal Groups

TL;DR

The paper addresses the structure of sequentially complete locally minimal Abelian groups, extending key weight-equality results from minimal groups to the locally minimal precompact setting and identifying when connected components coincide with those of completions. It introduces the notions of critical locally minimal groups and the compact completion class ${\frak C}_{Clm}$, and develops a framework based on delta-subgroups, co-NSS subgroups, and weak essentiality to analyze when local minimality is preserved under quotients. The authors prove that for locally precompact, sequentially complete, locally minimal Abelian groups, the weight of the connected component matches that of the completion, and under non-Ulam-measurable weights, the connected components themselves coincide, implying compactness. They then construct and characterize large families of critical locally minimal groups, describe the connected components of finite-dimensional members of ${\frak C}_{Clm}$, and provide both sufficient and necessary conditions delineating when a compact completion belongs to ${\frak C}_{Clm}$. The results highlight a stark contrast with the non-Abelian case, where analogous statements fail, and open several avenues regarding featheredness, Čech completeness, and extensions to broader classes of 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 $\CCC$.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem A
• Corollary A
• Definition 1.2
• Theorem B
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof