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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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I think locally indicable generally means that every finitely generated subgroup surjects onto the integers. –  Mustafa Gokhan Benli Oct 15 '11 at 23:05
    
Yes! You are right Mustafa! For every finitely generated subgroup of G there is a non trivial homomorphism to (R,+). –  Mahdi Teymuri Garakani Oct 15 '11 at 23:21

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