$\aleph_1$-free abelian non-Archimedean Polish groups

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 $\mathbb{Z}^\omega$; more strongly, $\mathbb{Z}^\omega$ 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.