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Let $G_1,G_2$ be two delta-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 '13 at 20:43
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See this paper or this paper. This result can be also proven using the arguments of Eleanor Rieffel here which avoid asymptotic cones (in the case you do not accept the Axiom of Choice).

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