Minimal pseudocompact group topologies on free abelian groups

AbstractA Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {σn:n∈N} of cardinals such thatw(G)=sup{σn:n∈N}andsup{2σn:n∈N}⩽|G|⩽2w(G), where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G|=2σ for some cardinal σ, or w(G)=min{σ:|G|⩽2σ}; moreover, the equality |G|=2w(G) holds whenever cf(w(G))>ω.For a cardinal κ, we denote by Fκ the free abelian group with κ many generators. If Fκ admits a pseudocompact group topology, then κ⩾c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on Fc is equivalent to the Lusin's Hypothesis 2ω1=c. For κ>c, we prove that Fκ admits a (zero-dimensional) minimal pseudocompact group topology if and only if Fκ has both a minimal group topology and a pseudocompact group topology. If κ>c, then Fκ admits a connected minimal pseudocompact group topology of weight σ if and only if κ=2σ. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology