Michael Puls
Published 2007-09-19, updated 2008-04-03Version 3
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 $\ell^p$-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 $\ell^p$-cohomology of $G$.
Comments: In the original paper the integers provide a counter example to Proposition 3.3. The reason is that I gave an incorrect definition for $p$-harmonic boundary. The new version has the correct defintion and some smaller changes
Group cohomology and $L^p$-cohomology of finitely generated groups
Graphs of bounded degree and the $p$-harmonic boundary
The ratio and generating function of cogrowth coefficients of finitely generated groups
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