${\aleph }_1$-free abelian non-Archimedean Polish groups

Abstract

An uncountable ${\aleph }_1$-free group cannot admit a Polish group topology but an uncountable ${\aleph }_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group ${\mathbb{Z}}^{\ddot{I}}‰$; more strongly, ${\mathbb{Z}}^{\ddot{I}}‰$ is separable. In this paper we investigate ${\aleph }_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so ${\aleph }_1$-free) abelian non-Archimedean Polish groups which are pairwise not topologically isomorphic. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_{\infty }$: separability, torsionlessness, ${\aleph }_1$-freeness and $\mathbb{Z}$-homogeneity.