Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

Abstract

In [9] we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper we show that our construction from [9] satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete, in fact, more strongly, we can ${\mathfrak{L}}_{{\omega }_1,\omega }$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}({\mathbb{Z}}^{(\omega )})$ is a complete analytic quasi-order.