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.
|Number of pages||25|
|Journal||Transactions of the American Mathematical Society|
|State||Published - 1 Jan 2017|
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics