Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups

Details Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups

2011-12-29

arxiv.org/abs/1112.6284v2

Mathematics

Metric Geometry

15 pages

Scientific paper

In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices $\mathds{Z}^n$ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.

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