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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.

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    $\begingroup$ Yes, this is proven in our papers with Leeb and Kleiner. $\endgroup$
    – Misha
    Nov 26, 2013 at 20:43

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See Kapovich and Leeb On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds, or Kapovich, Kleiner and Leeb Quasi-isometries and the de Rham decomposition. This result can be also proven using the arguments of Eleanor Rieffel in Groups quasi-isometric to $H^2\times R$ (Journal of the London Mathematical Society 64 Issue 1 (2001) 44–60 doi:10.1017/S0024610701002034) which avoid asymptotic cones (in the case you do not accept the Axiom of Choice).

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