Polynomials and harmonic functions on discrete groups

Tom Meyerovitch, Idan Perl, Matthew Tointon, Ariel Yadin

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsionfree nilpotent subgroup of finite index is always a polynomial in the Mal’cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos’s result using this notion of polynomials under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at most k is finitedimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree k surjectively onto the polynomials of degree k − 2. We then present some corollaries. In particular, we calculate precisely the dimension of the space of harmonic functions of polynomial growth of degree at most k on a virtually nilpotent group, extending an old result of Heilbronn for the abelian case, and refining a more recent result of Hua and Jost.

Original languageEnglish
Pages (from-to)2205-2229
Number of pages25
JournalTransactions of the American Mathematical Society
Volume369
Issue number3
DOIs
StatePublished - 1 Jan 2017

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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