Let's consider a (finitely generated) group $\Gamma$ and a coarse quotient map $q\colon\Gamma\to\mathbb{R}$. I'm interested in the 1-cocycle $\sigma\colon\Gamma\to\ell_\infty\Gamma$, defined by $\sigma(s):= q-s\cdot q$, where the $\Gamma$-action on $\ell_\infty\Gamma$ is given by $(s\cdot q)(x)=q(s^{-1}x)$. Let's assume that $\Gamma$ is amenable and take an (extremal) invariant mean $\nu$. Then $\ell_\infty\Gamma \subset L^2(\nu)$ and the $\Gamma$-action on $\ell_\infty\Gamma$ extends to a unitary $\Gamma$-action on $L^2(\nu)$. Thus we view $\sigma$ as a map into $L^2(\nu)$.
I would expect the answer to the following questions is true.
(1) Is $[\sigma]$ nonzero in the reduced cohomology $\overline{H^1}(\Gamma,L^2(\nu))$?
(2) Is the additive character $\nu\circ\sigma$ nonzero on $\Gamma$?
Obviously (2) implies (1) and the converse is true for $\Gamma$ with Shalom's property $\mathrm{H}_{\mathrm{T}}$. The answer to (1) is true if the coarse quotient map $q$ is given by the composition $q=\chi\circ\phi$ of a coarse isomorphism $\phi$ from $\Gamma$ into another group $\Lambda$ and an additive character $\chi\colon\Lambda\to\mathbb{R}$. See [Y. Shalom; Acta Math., 2004] (doi, zb, mr).
