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 $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups $Q_n(G,H) = I^{n-1}(G)I(H)/I^{n}(G)I(H)$, $H=G,N,T$, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with $N$ and $T$, in the following cases: for $n\le 4$ and arbitrary $G,N,T$ (except from one direct summand of $Q_4(G,N)$), and for all $n\ge 2$ if certain filtration quotients of $N$ and $T$ are torsionfree.Comment: 39 pages; paper thoroughly revised: notation and presentation improved, many details and new result added (Theorem 1.7

Polynomial functors from Algebras over a set-operad and non-linear Mackey functors

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations.Comment: 58 page

A seven-term exact sequence for the cohomology of a group extension

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).Comment: Updated version, new cocycle description adde