On the first $L^p$-cohomology of discrete groups

Abstract

For finitely generated groups $\Gamma$, the isomorphism between the first ${\ell }^p$-cohomology $H_{(p)}^1(\Gamma )$ and the reduced 1-cohomology with coefficients in ${\ell }^p(\Gamma )$ is exploited to obtain vanishing results for $H_{(p)}^1(\Gamma )$. The following cases are treated: groups acting on trees, groups with infinite center, wreath products, and lattices in product groups.