research

Graphs of bounded degree and the pp-harmonic boundary

Abstract

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

    Similar works

    Full text

    thumbnail-image