Let $G\_1$ and $G\_2$ be two finitely generated groups which are quasi-isometric in the sense of geometric group theory. 


Are their rational cohomology rings $H^{\ast}(G\_i; \mathbb Q)$ necessarily isomorphic? 



Do quasi-isometric groups have the same rational cohomology?
No. For example, $F\_2$ is quasi-isometric to $F\_3$ because the latter is finite index in the former, but their rational $H^1$s differ. 


