, Volume 129, Issue 1, pp 121-140

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

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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.

Oblatum 24-VI-1996 & 30-IX-1996