Comptes Rendus de l'Académie des Sciences. Série I, Mathématique vol:324 issue:3 pages:303-306
We know, from recent examples given by Benoist, Burde and Grunewald, that there exist polycyclic-by-finite groups Gamma, of rank h (the examples given were in fact nilpotent), admitting no properly discontinuous affine action on R-h. On the other hand for such Gamma's, it is always possible to construct a properly discontinuous smooth action of Gamma on R-h. Our main result is that any polycyclic-by-finite group Gamma of rank h acts properly discontinuously and by polynomial diffeomorphisms of bounded degree on R-h. This result follows from an unexpected and deep cohomology vanishing theorem. In this way, the best possible general answer to a question of John Milnor is probably given.