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 group G there exists a sequence {σn : n ∈ N} of cardinals such that w(G) = sup{σn : n ∈ N} and sup{2 σn : n ∈ N} ≤ |G| ≤ 2, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2 for some cardinal σ, or w(G) = min{σ : |G| ≤ 2}. Moreover, we show that the equality |G| = 2 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 2 ω1 = 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 = 2. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology. Throughout this paper all topological groups are Hausdorff. We denote by Z, P and N respectively the set of integers, the set of primes and the set of natural numbers. Moreover Q denotes the group of rationals and R the group of reals. For p ∈ P the symbol Zp is used for the group of p-adic integers. The symbol c stands for the cardinality of the continuum. For a topological group G the symbol w(G) stands for the weight of G, G̃ denotes the completion of G, the Pontryagin dual of a topological abelian group G is denoted by Ĝ. If H is a group and σ is a cardinal, then H(σ) is used to denote the direct sum of σ many copies of the group H. If G and H are groups, then a map f : G → H is called a monomorphism provided that f is both a group homomorphism and an injection. For undefined terms see [16, 17]. Definition 0.1. For a cardinal κ we use Fκ to denote the free abelian group with κ many generators.