  The existence of nonconstant harmonic Dirichlet functions on a Cayley graph
of a discrete group is equivalent to the nonvanishing of the first
L2-cohomology of the given group. It was first proven by Cheeger and Gromov
that such functions do not exists on the Cayley-graph of an amenable group. The
result was extended using Foster's averaging formula to transitive amenable
graphs by Medolla and Soardi. In this paper we extend this result further for
amenable graphs which are transitive in the category of quasi-isometries. Such
graphs are usually not quasi-isometric to transitive graphs. The extension
includes nets of amenable unimodular Lie-groups and geometrically amenable
homogeneous spaces.
