Polynomials and harmonic functions on discrete groups

TL;DR

This paper offers a simplified proof of a key result relating harmonic functions and polynomials on virtually nilpotent groups, extends dimension calculations, and refines existing theorems in the area.

Contribution

It provides a new proof of Alexopoulos's theorem using polynomial derivatives, extends dimension formulas for harmonic functions, and refines previous results for virtually nilpotent groups.

Findings

Simplified proof of harmonic polynomial characterization

Dimension formula for harmonic functions of polynomial growth

Refined results for virtually nilpotent groups

Abstract

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.