Polynomial alternatives for the group of affine motions

It is known that every polycyclic-by-finite group - even if it admits no af fine structure - allows a polynomial structure of bounded degree. A major o bstacle to a further development of the theory of these polynomial structur es is that the group of the polynomial diffeomorphisms of R-n, in contrast to the group of affine motions, is no longer a finite dimensional Lie group . In this paper we construct a family of (finite dimensional) Lie groups, e ven linear algebraic groups, of polynomial diffeomorphisms, which we call w eighted groups of polynomial diffeomorphisms. It tun-ls out that every poly cyclic-by-finite group admits a polynomial structure via these weighted gro ups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We int roduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existenc e of simply transitive actions of the corresponding Mal'cev completion. Thi s, and other properties, provide a strong analogy with the situation of aff ine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch. Mathematics Subject Classification (1991 ):57S20, 22E25.