Polynomials and harmonic functions on discrete groups

Autor: Matthew Tointon, Idan Perl, Ariel Yadin, Tom Meyerovitch
Rok vydání: 2016
Předmět:
Zdroj: Transactions of the American Mathematical Society. 369:2205-2229
ISSN: 1088-6850
0002-9947
DOI: 10.1090/tran/7050
Popis: Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free 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 k is finite-dimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree k k surjectively onto the polynomials of degree k − 2 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 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.
Databáze: OpenAIRE