Let $G\_1,G\_2$ be two Gromov-hyperbolic groups. If $G\_1\times \Bbb{Z}$ is quasi-isometric to $G\_2\times \Bbb{Z}$, must it be true that $G\_1$ is quasi-isometric to $G\_2$?


This is similar to the classic topology problem where crossing two non-homeomorphic spaces with $\Bbb{R}$ can create two homeomorphic spaces which are homeomorphic.



Quasi-isometric rigidity of certain products of groups
See Kapovich and Leeb *[On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds](http://www.math.ucdavis.edu/%7Ekapovich/EPR/KL_1995.pdf)*, or Kapovich, Kleiner and Leeb
*[Quasi-isometries and the de Rham decomposition](http://www.math.ucdavis.edu/%7Ekapovich/EPR/kkl.pdf)*. This result can be also proven using the arguments of Eleanor Rieffel in *[Groups quasi-isometric to $H^2\times R$](https://arxiv.org/abs/math/0005252)* (Journal of the London Mathematical Society **64** Issue 1 (2001) 44–60 doi:[10.1017/S0024610701002034](https://doi.org/10.1017/S0024610701002034)) which avoid asymptotic cones (in the case you do not accept the Axiom of Choice).


