  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.
