Polynomial structures for iterated central extensions of abelian-by-nilpotent groups

Abstract

A famous question of John Milnor [10] dealt with the problem of whether or not any torsion-free, polycyclic-by-finite group Γ occurs as the fundamental group of a compact, complete, affinely flat manifold. This is equivalent to saying that Γ admits a faithful affine representation, making it acting properly discontinuously on ℝ^K (K = Hirsch length of Γ), with compact quotient. This question was answered negatively by Y. Benoist [1] and D. Burde & F. Grunewald [2], even in the nilpotent case. However, it is known that any torsion-free, polycyclic-by-finite group Γ admits a smooth action on ℝ^K with compact quotient (see [3] and [7]). We might look at an affine mapping as being a polynomial of degree 1, while a smooth map, having in mind a power series expansion, can be regarded as “polynomial” of infinite degree. This paper investigates if it is possible to find anything in between those two. So our question becomes:

Does any (torsion-free) polycyclic-by-finite group Γ of rank K admit a properly discontinuous action on ℝ^K, expressed by polynomial functions, such that the quotient space is compact?

Research Assistant National Fund For Scientific Research (Belgium)