Graphs of bounded degree and the p-harmonic boundary
نویسنده
چکیده
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 l-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 and the first reduced l-cohomology of Γ.
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