Quadratic maps between non-abelian groups

The aim of this article is to study quadratic maps, in the sense of Leibman, between non-abelian groups. As a key result, we provide a complete characterization of quadratic maps through a universal construction that can be computed explicitly. Applications include a complete classification of quadratic maps on arbitrary abelian groups, proving the non-existence of polynomial maps of higher degree on perfect groups, establishing stability results for polynomial maps, and a $99\%$-inverse theorem for matrix-valued Gowers $U^k$-norms on perfect groups with bounded commutator width for arbitrary $k$.