The first $L^p$-cohomology of some finitely generated groups and   $p$-harmonic functions
  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. 
Michael J. Puls