Solvable Lie algebras, Lie groups and polynomial structures

In this paper, we study polynomial structures by starting on the Lie algebr a level, then passing to Lie groups to finally arrive at the polycyclic-by- finite group level. To be more precise, we first show how a general solvabl e Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Us ing this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on R-n which is polynomia l and of degree less than or equal to n(3). Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Gamma, which is of degree less than or equal to h(Gamma)(3) on almost the entire group (h (Gamma) being the Hirsch length of Gamma).