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

Abstract

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