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 $(<\aleph_0)$-tame but is $\aleph_0$-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.