POLYNOMIALITY PROPERTIES OF GROUP EXTENSIONS WITH A TORSION-FREE ABELIAN KERNEL

In studying torsion-free nilpotent groups of class 2 it is a key fact that each central group extension Z(n) --> E --> Z(m) is representable by a bilinear cocycle. So for investigating nilpotent groups of highe r class it is natural to ask for a generalization of this fact, namely when Z(m) is replaced by some torsion-free nilpotent group G and Z(n) by some torsion-free abelian group B with nilpotent G-action. In this paper we study a suitable notion of (bi)polynomial cocycles (in the s trong sense of polynomiality introduced by Passi) and prove the desire d representability theorem (4.3). This was known before only for centr al extensions with divisible kernel and with kernel Z if G is abelian, nilpotent of class 2 or if the quotients of the lower central series of G are torsion-free. Our representability result implies a convergen ce theorem for an approximation of H-2(G, B) by polynomial cohomology groups (4.4). Since the latter ones are well accessible to computation one obtains a formula for H-2(G, B) in terms of a presentation of G w hich can be evaluated by integral matrix calculus. More precisely, a g iven presentation of G amounts to a three-term cochain complex consist ing of finitely generated free Z-modules if the groups G and B are fin itely generated; the cohomology of this complex is identified with H-2 (G, B) in such a way that representing 2-cocycles are explicitly given in terms of integer valued rational polynomial functions (6.4). As an application, we establish an explicit bijection between torsion-free nilpotent groups and Z-torsion-free nilpotent Lie rings both finitely generated and nilpotent of class less than or equal to 3 (7.2). In cas e a given group extension is representable by a polynomial cocycle we also determine the minimal degree of polynomiality for which this hold s. Indeed, dropping the assumption that G is torsion-free nilpotent, w e give an intrinsic characterization of all group extensions with tors ion-free abelian kernel which are representable by a polynomial cocycl e of degree I n, provided that the cokernel is finitely generated and acts nilpotently on the kernel (4.2). Thus a polynomiality theory for group extensions as asked for by Passi is now achieved in case the ker nel is torsion-free. A motivation for this-apart from calculating H-2( G, B) explicitly-comes from a question posed by J. Milnor in 1977, nam ely whether all finitely generated, torsion-free virtually polycyclic groups arise as fundamental groups of compact, complete affinely flat manifolds. Even the case of torsion-free nilpotent groups is interesti ng but far from being understood, notably since counterexamples of thi s type have recently been discovered. A connection of Milnor's questio n for nilpotent groups with polynomial constructions was first indicat ed in the work of P. Igodt and K. B. Lee. Recently a very close connec tion of this type was established, and an obstruction theory developed for the problem in terms of polynomial cohomology. On the other hand, polynomiality properties of extensions with a torsion kernel are clos ely related to dimension subgroups, as was observed by Passi in the ca se of central extensions. An extension of this idea to the noncentral case is provided by Theorem 3.4 below. This might be of interest in co nnection with the recent discovery of Gupta and Kuz'min that the quoti ent group D-n(G)/gamma(n)(G) is an abelian normal but in general nonce ntral subgroup of G/gamma(n)(G). Moreover, we obtain a functorial equi valence between extensions of torsion-free nilpotent groups and of Z-t orsion-free nilpotent modules (5.1). This improves a general result of Reiner and Roggenkamp in the special situation of torsion-free nilpot ent groups. As applications, it yields an inductive cohomological desc ription of automorphism groups of torsion-free nilpotent groups [9] an d a generalization of the classical Dold-Kan equivalence-between simpl icial abelian groups and chain complexes of abelian groups-to simplici al groups of class 2. Applications in localization theory are also to be expected, namely to questions concerning the genus of torsion-free nilpotent groups. All the above-mentioned results are based on the mor e technical work of the first section. There, also, generalizations of theorems of Witt and Quillen are obtained concerning certain Lie alge bras associated with groups; these results might be of independent int erest. (C) 1996 Academic Press, Inc.