Polynomiality Properties of Group Extensions with a Torsion-Free Abelian Kernel

Abstract

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 bilinearcocycle. So for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact, namely when Z m is replaced by some torsion-free nilpotentgroup Gand Z n by some torsion-free abeliangroup Bwith nilpotent G-action. In this paper we study a suitable notion of ( bi) polynomialcocycles (in the strong sense of polynomiality introduced by Passi) and prove the desired representability theorem(4.3). This was known before only for centralextensions with divisiblekernel and with kernel Z if Gis abelian, nilpotent of class 2 or if the quotients of the lower central series of Gare torsion-free. Our representability result implies a convergence theoremfor 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 Gwhich can be evaluated by integral matrix calculus. More precisely, a given presentation of Gamounts to a three-term cochain complex consisting of finitely generated free Z -modules if the groups Gand Bare finitely 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 bijectionbetween torsion-free nilpotent groupsand Z -torsion-free nilpotent Lie ringsboth finitely generated and nilpotent of class≤3 (7.2). In case a given group extension is representable by a polynomial cocycle we also determine the minimal degreeof polynomiality for which this holds. Indeed, dropping the assumption that Gis torsion-free nilpotent, we give an intrinsic characterizationof all group extensions with torsion-free abelian kernel which are representable by a polynomial cocycle of degree ≤ 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 kernel 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, namely 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 interesting but far from being understood, notably since counterexamples of this type have recently been discovered. A connection of Milnor's question for nilpotent groups with polynomial constructions was first indicated in the work of P. Igodt and K. B. Lee. Recently a very close connection of this type was established, and an obstruction theorydeveloped for the problem in terms of polynomial cohomology. On the other hand, polynomiality properties of extensions with a torsionkernel are closely related to dimension subgroups, as was observed by Passi in the case of centralextensions. An extension of this idea to the noncentralcase is provided by Theorem 3.4 below. This might be of interest in connection with the recent discovery of Gupta and Kuz'min that the quotient group D n ( G)/γ n ( G) is an abelian normal but in general noncentral subgroup of G/γ n ( G). Moreover, we obtain a functorial equivalencebetween extensions of torsion-free nilpotent groupsand of Z - torsion-free nilpotent modules(5.1). This improves a general result of Reiner and Roggenkamp in the special situation of torsion-free nilpotent groups. As applications, it yields an inductive cohomological description of automorphism groupsof torsion-free nilpotent groups [9] and a generalization of the classical Dold-Kan equivalence—between simplicial abelian groups and chain complexes of abelian groups—to simplicial 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 more technical work of the first section. There, also, generalizations of theorems of Witt and Quillen are obtained concerning certain Lie algebras associated with groups; these results might be of independent interest.