In 1986, this reviewer named an abelian group \(G\) coslender, if, for every homomorphism \(f:G\to\prod\mathbb{Z}\) of \(G\) into a countable product of the integers, the projections of the image of \(f\) are \(0\), at all, except possibly finitely many coordinates [the reviewer, Glas. Mat., III. Ser. 21(41), 327–329 (1986; Zbl 0618.20038)]. This is equivalent to saying that the homomorphism group from \(G\) into \(\mathbb{Z}\) vanishes: \(\mathrm{Hom}(G,\mathbb{Z})=0\), equivalently if \(G\) has no direct summands isomorphic to \(\mathbb{Z}\); this places a large class of indecomposable groups (minus \(\mathbb{Z}\)) as a subclass of the class of coslender groups. In [Pitman Res. Notes Math. Ser. 204, 41–50 (1989; Zbl 0671.20053)], the reviewer exhibits an example of an \(\aleph_1\)-free slender and coslender group of cardinality \(2^{\aleph_0}\). In a follow up note [the reviewer and B. Goldsmith, Glas. Mat., III. Ser. 23(43), No. 2, 241–246 (1988; Zbl 0674.20032)], it was shown that any sum of coslender subgroups is again coslender and that a pure subgroup of a coslender group need not be coslender, even in the finite rank case. It was shown that there are superdecomposable coslender groups of any cardinality. It was also pointed out that (under CH) there are \(\aleph_1\)-free coslender groups of cardinality \(\aleph_1\).
The author of the paper under review proved in [the author, Cubo 9, No. 2, 59–79 (2007; Zbl 1144.03034)] that there are \(\aleph_n\)-free coslender abelian groups. In this paper the author sets the goal of finding the supremum of \(\lambda\), such that there are coslender \(\lambda\)-free groups. To that end, he proves that there are \(\aleph_{\omega_1\cdot n}\)-free (dubbed “quite free”) coslender groups (which are not Whitehead) and that is the farthest in \(\aleph\)-freeness of coslender groups that one can go. Namely he proves that, \(\kappa=\aleph_{\omega_1\cdot\omega}\) is the first cardinal for which it cannot be proved within ZFC that there are \(\kappa\)-free coslender groups. On the other hand, under the assumption that there are countably many supercompact cardinals in the ground model, the following statement is forced: Every non-trivial \(\kappa\)-free abelian group is not coslender, in order to show that it is consistent that every \(\kappa\)-free abelian group is not coslender. One of the difficulties of this paper is in that cardinality and freeness differ. The author uses his favorite method of “quite-free” (\(n\)-dimensional) black boxes to arrive at a number of the results. When generalizing from abelian groups to modules, the author assumes module category where pure submodules of free modules are again free (modules over “hereditary domains”). It may be that proving the existence of \(\lambda\)-free coslender groups need not require such heavy machinery as employed in the paper, however the techniques developed in the paper may have other applications.