THE p-HARMONIC BOUNDARY FOR FINITELY GENERATED GROUPS AND THE FIRST REDUCED l-COHOMOLOGY

Let p be a real number greater than one and let G be a finitely generated, infinite group. In this paper we introduce the p-harmonic boundary of G. We then characterize the vanishing of the first reduced l-cohomology of G in terms of the cardinality of this boundary. Some properties of p-harmonic boundaries that are preserved under rough isometries are also given. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on G, the p-harmonic boundary of G and the first reduced l-cohomology of G.