Polynomial structures for nilpotent groups

Abstract:

If a polycyclic-by-finite rank-$K$ group $\Gamma$ admits a faithful affine representation making it acting on $\mathbb {R}{K}$ 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.