Examples of non-tame abstract elementary classes of abelian groups

We construct an abstract elementary class $K_1$ of torsion-free abelian groups such that $K_1$ is not $(<{\mathrm{\aleph }}_0)$-tame but is ${\mathrm{\aleph }}_0$-tame. This answers a question of [BoVa17]. Furthermore, for every regular uncountable cardinal $\mu$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^{\mu })$ of torsion-free abelian groups such that $K_2(2^{\mu })$ is not $(<\mu )$-tame. $K_1$ and $K_2(2^{\mu })$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language.