On Fox and augmentation quotients of semidirect products

Let G be a group which is the semidirect product of a normal subgroup N and some subgroup T . Let In(G), n ≥ 1, denote the powers of the augmentation ideal I(G) of the group ring Z(G). Using homological methods the groups Qn(G,H) = In−1(G)I(H)/In(G)I(H), H = G,N, T , are functorially expressed in terms of enveloping algebras of certain Lie rings associated with N and T , in the following cases: for n ≤ 4 and arbitrary G,N, T (except from one direct summand of Q4(G,N)), and for all n ≥ 2 if certain filtration quotients of N and T are torsionfree. Introduction. The group ring Z(G) of a group G is naturally filtered by the powers I(G), n ≥ 1, of its augmentation ideal I(G). It is a long-studied problem to determine the so-called augmentation quotients Qn(G) = I (G)/I(G) in terms of the structure of G , also because of their close link with the dimension subgroups Dn(G) = G ∩ (1 + I(G)) which can be inductively described as Dn+1(G) = Ker(Dn(G)→ Qn(G)). The groups Qn(G) were determined for n = 2 by Passi [21], Sandling [24] and Losey [19] for abelian, finite and finitely generated groups G and for n = 3, 4 and finite G by Tahara [27], [28]; a functorial description for all groups was given for n = 2 by Bachmann and Gruenenfelder [2] and for n = 3 in [3], based on Quillen’s approximation of the graded ring Gr(Z(G)) = Z ⊕ ⊕ n≥1Qn(G) by the enveloping ring of the Lie ring of G , see [23], [22] or section 1 below. More generally, the investigation of the classical Fox