The classification problem for extensions of torsion-free abelian groups, I

Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation ${\mathcal{R}}_{\mathbf{E}\mathbf{x}\mathbf{t}(C,A)}$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra.