In this talk, I discuss the potential Borel complexity of the isomorphism relation for short exact sequences of countable torsion-free abelian groups. For this, we use both set-theoretic methods (in particular the theory of groups with a Polish cover and the notion of Solecki subgroups) and some categorical tools. One of the results is that for a certain class of groups \(A\) we can find \(C\) such that the classification problem corresponding to \(\mathbf{Ext}(C,A)\) can have arbitrarily high potential Borel complexity. I will also present some results on the low-complexity cases and, time permitting, discuss the problem in the case of rigid groups.
This is a work in progress with Martino Lupini.