The first $L^p$-cohomology of some groups with one end

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.