Abstract
A mapping ϕ of a groupG to a groupF is said to be polynomial if it trivializes after several consecutive applications of operatorsD h ,h ∈G, defined byD h ϕ(g)=ϕ(g) −1 ϕ(gh). We study polynomial mappings of groups, mainly to nilpotent groups. In particular, we prove that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any polynomial mappingG→F to a nilpotent groupF splits into a homomorphismG→G’ to a nilpotent groupG’ and a polynomial mappingG’→F. We apply the obtained results to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group.
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This work was supported by NSF, Grants DMS-9706057 and 0070566.
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Leibman, A. Polynomial mappings of groups. Isr. J. Math. 129, 29–60 (2002). https://doi.org/10.1007/BF02773152
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DOI: https://doi.org/10.1007/BF02773152