On the Cayley graph of an amenable group
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.
scientific article; zbMATH DE number 1354992