Cohomology of Locally Indicable groups

Question

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?