Let $p$ be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first $L^p$-cohomology space of some groups that have one end. We also make a connection between the first $L^p$-cohomology space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number $p$ such that the first $L^p$-cohomology space of a nonelementary hyperbolic group does not vanish.
The first $L^p$-cohomology of some finitely generated groups and $p$-harmonic functions
The p-harmonic boundary for finitely generated groups and the first reduced \ell_p-cohomology
Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree
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