18.111 (2014)

Let $G$ be a discrete countable group, given as a central extension $0 \to \mathbb{Z} \to G \to Q \to 0$. Assume that either $G$ is quasi-isometric to $\mathbb{Z} \times Q$, or that $\mathbb{Z} \to G$ is a quasi-isometric embedding. Does that imply that $G$ comes from a bounded cocycle on $Q$?