Quadratic maps between non-abelian groups

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a $1\mathrm{\%}$-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this article, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the $99\mathrm{\%}$ regime for the $U^k$-norm on perfect groups of bounded commutator width. This analysis prompts a classification of Leibman’s quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.