Harmonic functions with finite <i>p</i>-energy on lamplighter graphs are constant

Antoine Gournay

doi:10.1016/j.crma.2016.04.015

Group theory/Combinatorics

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

Presented by: the Editorial Board

Antoine Gournay ¹

¹ TU Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 762-765.

Abstract
Résumé

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 $ℓ^{p}$ (i.e. finite p-energy) for any $p \in [1, \infty [$ except constants (and, equivalently, that their reduced $ℓ^{p}$ cohomology is trivial in degree one). Similar arguments are then applied to many direct products of graphs to conclude the same (including all direct products of Cayley graphs). The proof relies on a theorem of Thomassen [16] on spanning lines in squares of graphs.

Le but de cette note est de montrer que plusieurs graphes d'allumeurs de réverbères, où le graphe d'espace est infini avec au plus deux bouts et le graphe des lampes a au plus deux bouts, ne possèdent pas de fonction harmonique non constante à gradient $ℓ^{p}$ (i.e. une p-energie finie) qu'importe le $p \in [1, \infty [$ (et, de manière équivalente, que leur cohomologie $ℓ^{p}$ réduite est triviale en degré un). Des arguments similaires permettent aussi de conclure pour plusieurs produits directs de graphes (y compris tous les graphes de Cayley). Les démonstrations reposent sur un théorème de Thomassen [16] sur les lignes couvrantes dans le carré des graphes.

Received: 2015-12-07
Accepted: 2016-04-18
Published online: 2016-05-25

DOI: 10.1016/j.crma.2016.04.015

Author's affiliations:

Antoine Gournay ¹

¹ TU Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany

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Antoine Gournay. Harmonic functions with finite p-energy on lamplighter graphs are constant. Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 762-765. doi : 10.1016/j.crma.2016.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.04.015/

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