Freely indecomposable almost free groups with free abelianization

For certain uncountable cardinals $\kappa$ we produce a group of cardinality $\kappa$ which is freely indecomposable, strongly $\kappa$-free, and whose abelianization is free abelian of rank $\kappa$. The construction takes place in G"odel’s constructible universe $L$. This strengthens an earlier result of Eklof and Mekler.