Quadratic maps between non-abelian groups

2024/12/19 by Jamneshan, Asgar, Thom, Andreas · 1 citation
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2412.14908

Abstract

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%-inverse theorem for the U²-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% 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.

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