Harmonic functions with finite $p$-energy on lamplighter graphs are constant

TL;DR

This paper proves that certain lamplighter and product graphs do not admit non-constant harmonic functions with finite $p$-energy, implying trivial first reduced $ ext{l}^p$ cohomology, using spanning line theorems.

Contribution

It establishes the non-existence of non-constant harmonic functions with finite $p$-energy on specific lamplighter and product graphs, extending previous cohomological results.

Findings

Lamplighter graphs with specific properties admit only constant harmonic functions with finite $p$-energy.

Many direct product graphs, including Cayley graphs, do not admit non-constant harmonic functions with $ ext{l}^p$ gradients.

The proof employs Thomassen's theorem on spanning lines in graph squares.

Abstract

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in $\ell^p$ (\ie finite $p$-energy) for any $p\in [1,\infty[$ except constants (and, equivalently, that their reduced $\ell^p$ cohomology is trivial in degree one). Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in $\ell^p$. The proof relies on a theorem of Thomassen on spanning lines in squares of graphs.