Polynomial mappings of groups
- A. Leibman
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- [B] A. Babakhanian,Cohomological Methods in Group Theory, Marcel Dekker, New York, 1972.
- [BL] V. Bergelson and A. Leibman,A nilpotent Roth theorem, to appear in Inventiones Mathematicae.
- [D] H. A. Dye,On the ergodic mixing theorem, Transactions of the American Mathematical Society118 (1965), 123–130. CrossRef
- [F] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981.
- [KM] M. Kargapolov and Ju. Merzljakov,Fundamentals of the Theory of Groups, Springer-Verlag, Berlin, 1979.
- [L1] A. Leibman,Polynomial sequences in groups, Journal of Algebra201 (1998), 189–206. CrossRef
- [L2] A. Leibman,Multiple recurrence theorem for measure preserving actions of a nilpotent group, Geometric and Functional Analysis8 (1998), 853–931. CrossRef
- [L3] A. Leibman,Structure of unitary actions of finitely generated nilpotent groups, Ergodic Theory and Dynamical Systems20 (2000), 809–820. CrossRef
- [Sz] E. Szemerédi,On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica27 (1975), 199–245.
- Polynomial mappings of groups
Israel Journal of Mathematics
Volume 129, Issue 1 , pp 29-60
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors
- A. Leibman (1)
- Author Affiliations
- 1. Department of Mathematics, The Ohio State University, 43210-1174, Columbus, OH, USA