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arXiv · 2412.14908

Quadratic maps between non-abelian groups

Asgar Jamneshan·Andreas Thom
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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 $U^2$-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 $99\%$ 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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Asgar Jamneshan, Andreas Thom. 2025-08-29. Quadratic maps between non-abelian groups. https://doi.org/10.1017/s0305004126101911

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