Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion Abelian groups. (Q1887445)

Let \(\mathbb{Z} G\) be the integral group ring of the group \(G\) and let \(\Delta(G)\) be the augmentation ideal of \(\mathbb{Z} G\), i.e. \(\Delta(G)\) is the free Abelian group on the elements \(g-1\) (\(g\in G\)). Then the \(n\)-th power \(\Delta^n(G)\) of \(\Delta(G)\) is generated as an Abelian group by the standard generators \((g_1-1)(g_2-1)\cdots(g_n-1)\) with \(g_1,g_2,\dots,g_n\in G\). It is a classical problem in the theory of group rings to find all relations among the standard generators of \(\Delta^n(G)\). The authors solve this problem when \(G\) is a torsion free or torsion Abelian group. The results are applied to describe the homology of the sequence \(\Delta^n(N)G\to\Delta^n(G)\to\Delta^n(G/N)\), where \(G\) is an Abelian group and \(N\subseteq G\) is a subgroup.