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

TL;DR

The paper strengthens the descriptive-set-theoretic profile of countable torsion-free abelian groups by proving that $\mathrm{TFAB}_\omega$ is faithfully Borel complete and that graphs can be interpreted in $\mathrm{TFAB}_\omega$ via $\mathfrak{L}_{\omega_1,\omega}$-interpretations. It introduces a two-part construction—a combinatorial frame together with a universal abelian-group realization—that yields Borel reductions and allows the interpretation of graph structures within TFAB_ω. It further shows that the pure embeddability relation on countable TFAB_ω is a complete analytic quasi-order, and that elementary embeddability among countable models of $Th(\mathbb{Z}^{(\omega)})$ shares the same analytic complexity. These results connect the complexity of graph embeddings to the structure theory of countable torsion-free abelian groups, with implications for anti-classification questions and Vaught-type considerations in this algebraic context.

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 $\mathfrak{L}_{ω_1, ω}$-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (equiv., elementary embeddability) among countable models of $\mathrm{Th}(\mathbb{Z}^{(ω)})$ is a complete analytic quasi-order.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Corollary 1.4
• Definition 1.5
• Definition 2.1
• Definition 3.3
• Claim 3.4
• proof
• proof : Proof of Theorem \ref{['first_theorem']}