Let p be a real number greater than one and let G be a connected graph of
bounded degree. In this paper we introduce the p-harmonic boundary of G. We
use this boundary to characterize the graphs G for which the constant
functions are the only p-harmonic functions on G. It is shown that any
continuous function on the p-harmonic boundary of G can be extended to a
function that is p-harmonic on G. Some properties of this boundary that are
preserved under rough-isometries are also given. Now let Γ be a finitely
generated group. As an application of our results we characterize the vanishing
of the first reduced ℓp-cohomology of Γ in terms of the
cardinality of its p-harmonic boundary. We also study the relationship
between translation invariant linear functionals on a certain difference space
of functions on Γ, the p-harmonic boundary of Γ with the first
reduced ℓp-cohomology of Γ.Comment: Give a new proof for theorem 4.7. Change the style of the text in the
first two section