Polynomial Growth Harmonic Functions on Groups of Polynomial Volume Growth

TL;DR

This paper establishes an optimal polynomial bound on the dimension of harmonic functions with polynomial growth on Cayley graphs of groups with polynomial volume growth, extending known Riemannian results to a broader graph setting.

Contribution

It provides the first sharp estimate for the dimension of polynomial growth harmonic functions on groups of polynomial volume growth, including graphs roughly isometric to such groups.

Findings

Dimension of harmonic functions is at most proportional to d^{D-1}.

The estimate is polynomial in the growth degree d.

Results extend to graphs roughly isometric to Cayley graphs.

Abstract

We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions. More precisely, the dimension of this space of harmonic functions is at most of order $d^{D-1}$. As in the already known Riemannian case, this estimate is polynomial in the growth degree. More generally, our techniques also apply to graphs roughly isometric to Cayley graphs of groups of polynomial volume growth.