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Polynomials and Harmonic Functions on Discrete Groups by Meyerovitch, Tom; Perl, Idan; Tointon, Matthew; Yadin, Ariel is a scholarly article available to read on EtoBox.

What is Polynomials and Harmonic Functions on Discrete Groups about?

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 is finite dimensional. 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.

Author

Meyerovitch, Tom; Perl, Idan; Tointon, Matthew; Yadin, Ariel

Published

2015

Language

EN