The Structure of Sequentially Complete Locally Minimal Groups

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 $\CCC$.