Polycyclic-by-finite groups admit a bounded-degree polynomial structure

Abstract

For a polycyclic-by-finite group $\Gamma$ , of Hirsch length $h$ , an affine (resp. polynomial) structure is a representation of $\Gamma$ into ${\rm Aff}({\Bbb R}^{h})$ (resp. ${\rm P}({\Bbb R}^h)$ , the group of polynomial diffeomorphisms) letting $\Gamma$ act properly discontinuously on ${\Bbb R}^{h}$ . Recently it was shown by counter-examples that there exist groups $\Gamma$ (even nilpotent ones) which do not admit an affine structure, thus giving a negative answer to a long-standing question of John Milnor. We prove that every polycyclic-by-finite group $\Gamma$ admits a polynomial structure, which moreover appears to be of a special (“simple”) type (called ”canonical”) and, as a consequence of this, consists entirely of polynomials of a bounded degree. The construction of this polynomial structure is a special case of an iterated Seifert Fiber Space construction, which can be achieved here because of a very strong and surprising cohomology vanishing theorem.