, Volume 129, Issue 1, pp 29-60

Polynomial mappings of groups

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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 ,hG, 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 mappingGF to a nilpotent groupF splits into a homomorphismGG’ 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.

This work was supported by NSF, Grants DMS-9706057 and 0070566.