Polycyclic-by-finite groups admit a bounded-degree polynomial structure (Q678260)

For a polycyclic-by-finite group \(\Gamma\), of Hirsch length \(h\), an affine (resp. polynomial) structure is a representation of \(\Gamma\) into \(\text{Aff}(\mathbb{R}^h)\) (resp. \(P(\mathbb{R}^h)\), the group of polynomial diffeomorphisms) letting \(\Gamma\) act properly discontinuous on \(\mathbb{R}^h\). 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.