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.

15 pages, to appear in Ann. Global Anal. Geom