Minimal pseudocompact group topologies on free abelian groups

Abstract

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 {σ_n : n\in N} of cardinals such that w(G) = sup {σ_n : n \in N} and sup {2^{σ_n} : n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^σfor some cardinal σ, or w(G) = min {σ: |G| \leq 2^σ}; moreover, the equality |G| = 2^{w(G)} holds whenever cf (w(G)) > ω. For a cardinal κ, we denote by F_κthe free abelian group with κmany generators. If F_κadmits a pseudocompact group topology, then κ\geq c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F_c is equivalent to the Lusin's Hypothesis 2^{ω_1} = c. For κ> c, we prove that F_κadmits a (zero-dimensional) minimal pseudocompact group topology if and only if F_κhas both a minimal group topology and a pseudocompact group topology. If κ> c, then F_κadmits a connected minimal pseudocompact group topology of weight σif and only if κ= 2^σ. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.