Quadratic maps between groups

The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi’s polynomial maps and groups of degree 2 is established and used to study the structure of the latter. Introduction. Polynomial maps appear in nilpotent group theory for a long time, originally in the form of rational (numerical) functions, for example in the Hall-Petrescu formula or the group law of torsionfree nilpotent groups when written with respect to a Mal’cev basis. An intrinsic notion of polynomial maps from groups to abelian groups was introduced by Passi [30], together with a universal example G → Pn(G) where the abelian group Pn(G) is called “polynomial group”. Passi’s motivation came from the study of dimension subgroups; since then, his construction turned out to provide a key tool in the study of many other problems: in the theory of group schemes [7] as well as in the theory of nilpotent groups, concerning their second (co)homology [15], [16], automorphism groups or simplicial objects [13], [17]. However, a need to study polynomial maps between arbitrary groups comes from unstable homotopy theory; after Baues’ [3] and the author’s [12] study of metastable homotopy groups and Moore spaces [4] the foundations of “quadratic algebra” were layed in [5] where a notion of quadratic maps with nonabelian target group first appeared. Since then, in the steadily growing literature on quadratic algebra and its applications, various variants and properties were exhibited when needed, in work of Baues, Jibladze, Muro, Pirashvili and the author