Minimal pseudocompact group topologies on free abelian groups

A 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} and sup {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.