Cuts, flows and gradient conditions on harmonic functions

TL;DR

This paper investigates harmonic functions with specific gradient conditions on Cayley graphs of certain groups, showing their non-existence in various group structures, which advances understanding of harmonic analysis on groups.

Contribution

It establishes the non-existence of harmonic functions with gradient in c_0 or ℓ^p on Cayley graphs of particular infinite and metabelian groups, linking group structure to harmonic function properties.

Findings

No harmonic functions with gradient in c_0 on Cayley graphs of certain infinite groups.

No harmonic functions with gradient in ℓ^p on Cayley graphs of metabelian groups.

Group structure influences the existence of harmonic functions with specific gradient conditions.

Abstract

Reduced cohomology motivates to look at harmonic functions which satisfy certain gradient conditions. If $G$ is a direct product of two infinite groups or a (FC-central)-by-cyclic group, then there are no harmonic functions with gradient in $c_0$ on its Cayley graphs. From this, it follows that a metabelian group $G$ has no harmonic functions with gradient in $\ell^p$.