The structure of sequentially complete locally minimal groups

Generalizing results of Dikranjan and Tkachenko (2001) and Dikranjan and Uspienskij (2023) 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 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 CClm 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 CClm.