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Let $G$ be a locally indicable group (i.e. there is a non trivial homomorphism from $G$ to the real additive group $(\mathbb{R},+)$) and $l^2(G)$ be the Hilbert space with the base $G$. Is it true that $H^1(G,l^2(G))$ does not vanish?

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    Yes! You are right Mustafa! For every finitely generated subgroup of G there is a non trivial homomorphism to (R,+).2011-10-15

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