Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first $L^p$-cohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups.
The first $L^p$-cohomology of some groups with one end
The p-harmonic boundary for finitely generated groups and the first reduced \ell_p-cohomology
Group cohomology and $L^p$-cohomology of finitely generated groups
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