On the Cayley graph of an amenable group
type: Item
instance of: scholarly article
title: On the Cayley graph of an amenable group
zbMATH Open document ID: 0924.05033
DOI: 10.1023/A:1006511917850
published in: Acta Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated infinite group with generator set \(S\) and let \(X\) be the oriented Cayley graph of \(\Gamma\) associated to the generator set \(S\). Further let \(B_\Gamma\) be the Hilbert space of all square-integrable functions on the edges of \(X\) obeying the Kirchhoff equations. The combinatorial Laplacian and harmonic 1-cochains are defined in standard way. The following theorem is proven: If \(\Gamma\) is amenable, then the space of compactly supported harmonic 1-cochains on \(X\) is dense in the space of all square-integrable harmonic 1-cochains.
reviewed by: Piotr Kosiński
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 46E99
zbMATH DE Number: 1354992
zbMATH Keywords: infinite groups
zbMATH Keywords: Cayley graph
zbMATH Keywords: harmonic 1-cochains
zbMATH Keywords: Hilbert space
zbMATH Keywords: Kirchhoff equations
zbMATH Keywords: Laplacian
author: Gábor Elek
MaRDI profile type: Publication
full work available at URL: a:1006511917850
OpenAlex ID: W1658971025
description: scientific article; zbMATH DE number 1354992