There is some question from geometric group theory: One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
- $\Gamma$ and $\Lambda$ are quasi-isometric.
- The Mal'cev completion of $\Gamma$ is isomorphic to the Mal'cev completion of $\Lambda$.
I am wondering in what direction this question is most useful. Is it easier to find the Mal'cev completion of these groups rather than proving these are quasi-isometric? Is it the other way around? Or is there no clear answer (ie. both techniques could be useful depending on the situation)?
Thank you for your effort/answer.