Recent advances in minimal topological groups

AbstractThis survey presents some recent trends and results (most of them unpublished) in minimal groups. The following are the main direction: 1.(a) permanence properties of minimal groups;2.(b) complete minimal groups;3.(c) countably compact minimal groups;4.(d) algebraic structure of minimal abelian groups.In (a) we discuss preservation of minimality under the main group-theoretic operations: taking (direct or semidirect) products, quotients and (dense or closed) subgroups. Particular emphasis is given to infinite products and the critical power of minimality of a minimal abelian group G (this is the least nonminimal power of G provided such powers exist, otherwise κ(G) = 1). In particular, there exist (strongly) pseudocompact minimal abelian groups G with κ(G) = ω1, while for countably compact minimal abelian groups G either κ(G) = 1, when the connected component of G is compact, or κ(G) = ω; otherwise.(b) and (c) are parallel but in opposite direction. In (b) we put completeness-like conditions on the minimal groups and see when they are compact. In (c) we impose countable compactness on the minimal groups and look for further conditions which may yield compactness. In this way (b) becomes a chase for precompactness, while (c) becomes a chase for completeness. This is why we dedicate in (b) special attention to the celebrated precompactness theorem for minimal abelian groups of Prodanov and Stoyanov and we offer some examples and comments in the case of nilpotent groups. Here we consider also stronger completeness conditions, as local compactness, completeness of all quotients, etc.It turns out that the question whether connected, countably compact, minimal abelian groups are compact depends on the existence of measurable cardinals. More precisely, the connected component of a countably compact minimal abelian group G must be compact whenever its size is not Ulam-measurable. In such a case κ(G) = 1 and the algebraic structure of G can be completely described. Under the assumption that there exist measurable cardinals one can construct a noncompact ω-bounded, connected, minimal abelian group G (this entails, of course, κ(G) = ω).In (d) we give an alternative exposition of the known results on this question. Our approach takes into account the connection between the algebraic invariants of the group and its topological properties. We pay special attention to the case of abelian groups of free-rank < c resolved by Schinkel (1990, Dissertation) and answer an open question of his regarding the case of torsion-free groups of large free-ranks. We show that ZFC cannot answer the question whether the free abelian group F of rank c admits minimal pseudocompact group topologies, even if F admits both (totally) minimal group topologies and pseudocompact group topologies