Quadratic maps between non-abelian groups
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915901867032576 |
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| author | Jamneshan, Asgar Thom, Andreas |
| author_facet | Jamneshan, Asgar Thom, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic maps between non-abelian groups Jamneshan, Asgar Thom, Andreas Group Theory 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. |
| title | Quadratic maps between non-abelian groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2412.14908 |