Let \(\mathbb{Z} G\) be the integral group ring of a group \(G\). If \(G\) is a finite group, then the author [F. E. A. Johnson, “Stable modules and the \(D(2)\)-problem” (Lond. Math. Soc. Lect. Note Ser. 301), CUP (2003; Zbl 1055.57002)] has studied a class \(\Omega_1(\mathbb{Z})\) of \(\mathbb{Z} G\)-modules that are stably equivalent to the augmentation ideal of \(\mathbb{Z} G\). In this paper, the author continues these investigations for finitely generated infinite groups \(G\) satisfying the condition \(\text{Ext}^1(\mathbb{Z},\mathbb{Z} G)=0\).
For a given group \(G\) let \(\mathbf{SF}\) be the tree of nonzero stably free \(\mathbb{Z} G\)-modules. The main result asserts that the correspondence \(J\to\operatorname{Hom}_{\mathbb{Z} G}(J,\mathbb{Z} G)\) defines a surjective level preserving mapping of trees \(\delta\colon\Omega_1(\mathbb{Z})\to\mathbf{SF}\). Moreover, \(\delta\) induces a one-to-one correspondence between the minimal level of \(\Omega_1(\mathbb{Z})\) and the isomorphism classes of stably free modules of rank 1.