Polynomials and harmonic functions on discrete groups
Autor: | Matthew Tointon, Idan Perl, Ariel Yadin, Tom Meyerovitch |
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Rok vydání: | 2016 |
Předmět: |
Pure mathematics
Polynomial Degree (graph theory) Applied Mathematics General Mathematics 010102 general mathematics 01 natural sciences Nilpotent Harmonic function 0103 physical sciences Finitely generated group 0101 mathematics Abelian group Nilpotent group 010306 general physics Laplace operator Mathematics |
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 |
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