Transactions of the american mathematical society vol:348 issue:1 pages:77-97
If a polycyclic-by-finite rank-K group Gamma admits a faithful affine representation making it acting on R(K) properly discontinuously and with compact quotient, we say that Gamma admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Gamma. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups N, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal'cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ''affine defect number''. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.