  Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$
denote the Banach space $\{\sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p <
\infty \}$. In this paper we will study the first cohomology group of $G$ with
coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$.
Most of our results will be for a class of groups that contains all finitely
generated, infinite nilpotent groups.

Functional Analysis;Group Theory