2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68853We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to Π} (X_0, μ_0)_g$ for $(X_{0},μ_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.9 pages; some corrections in Lemma 3.2 (Dec. 18, 2003), 10 pagesOperator AlgebrasGroup Theory46L10 (Primary) 22D40, 28D05, 28D15 (Secondary)text