On Fox Quotients of Arbitrary Group Algebras

Abstract

Certain subquotients of group algebras are determined as a basis for subsequent computations of relative Fox and dimension subgroups. More precisely, for a group G and N-series G of G let I n R,G (G), n ≥ 0, denote the filtration of the group algebra R(G) induced by G , and I R (G) its augmentation ideal. For subgroups H of G, left ideals J of R(H) and right H -submodules M of I Z Z (G) the quotients I R (G)J/M J are studied by homological methods, notably for M = I R (G

On Fox and augmentation quotients of semidirect products

“…As in [9], our arguments are most conveniently formulated in the language of pushouts of abelian groups, thereby using their elementary properties, in particular the gluing of pushouts and the link between the kernels of parallel maps in a pushout square, see [27]. Consider the following diagram whose top row is exact taking the right-hand map to be given by the projection to the cokernel of the left-hand map followed by the isomorphism D −1 H in (1.14), and where j 1 is induced by the corresponding injection and D is given by restriction of j 1 D H (3) .…”

“…As to the map θ G H n for n 3, part of its kernel is determined in [9] for H = γ ; indeed, all arguments there remain valid for arbitrary H as above, thus providing an explicitly defined subgroup…”

“…) is more complicated, see [9] for the case H = γ ; in the special case of interest here it is described in the next section. The structure of the related groups U …”

“…Recently, Passi and Mikhailov pass from homological to homotopical methods [22], [23]. In [9] and in this paper we combine enveloping algebras and new homological observations to study the more general quotients Q n (G, H) = I n−1 (G)I(H)/I n (G)I(H) for some subgroup H of G; we call these Fox quotients because of their close relation with the classical Fox subgroups G ∩ (1 + I n−1 (G)I(H)). Fox quotients (and some related groups, see [18], [11], [12]) were also extensively studied in the literature, but, except from [9], only under suitable splitting assumptions, in particular when H is a semidirect factor of G. In fact, Sandling's [29] and later Tahara's work [33] on augmentation quotients of semidirect products G = N ⋊ T had split the study of Fox quotients into two classes of independent problems: the study of certain filtration quotients of Z(N) and Z(T ) on the one hand and of product filtrations F n = ∆ n−i I i (T ) on the other hand where (∆ i ) i≥1 is one of two natural filtrations of Z(N), see section 1.…”

“…In [9] and in this paper we combine enveloping algebras and new homological observations to study the more general quotients Q n (G, H) = I n−1 (G)I(H)/I n (G)I(H) for some subgroup H of G; we call these Fox quotients because of their close relation with the classical Fox subgroups G ∩ (1 + I n−1 (G)I(H)). Fox quotients (and some related groups, see [18], [11], [12]) were also extensively studied in the literature, but, except from [9], only under suitable splitting assumptions, in particular when H is a semidirect factor of G. In fact, Sandling's [29] and later Tahara's work [33] on augmentation quotients of semidirect products G = N ⋊ T had split the study of Fox quotients into two classes of independent problems: the study of certain filtration quotients of Z(N) and Z(T ) on the one hand and of product filtrations F n = ∆ n−i I i (T ) on the other hand where (∆ i ) i≥1 is one of two natural filtrations of Z(N), see section 1. In a series of papers Khambadkone and later Karan and Vermani expressed the quotients of these product filtrations in terms of tensor products of the groups ∆ n−i /∆ n−i+1 and I i (T )/I i+1 (T ), for low values of n and under additional assumptions, assuming either G finite and N finitely generated or nilpotent [17], [18], [19], or assuming torsion-freeness of sufficiently many filtration quotients of N and T [13], [15], [14], [34].…”