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

TL;DR

This paper explores the landscape of uncountable $eth_1$-free abelian groups with non-Archimedean Polish topologies, revealing a rich diversity of such groups and classifying key properties as complex descriptive set-theoretic sets.

Contribution

It demonstrates the existence of continuum many non-isomorphic separable abelian non-Archimedean Polish groups and classifies properties like separability and torsionlessness as complete co-analytic sets.

Findings

Existence of continuum many pairwise non-isomorphic separable abelian non-Archimedean Polish groups.

Separable, torsionless, $eth_1$-free, and $bZ$-homogeneity properties are complete co-analytic sets.

Uncountable $eth_1$-free groups cannot admit Polish group topologies, but some can.

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 $\mathbb{Z}^\omega$; 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.