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Let $G$ be a discrete, finitely generated, and amenable group. Let $H$ be a group which is quasi- isometric to $G$. Is $H$ amenable?

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Yes. This is essentially an immediate consequence of the Folner sets definition of amenability. You can find references at Theorem 10.23 of the article by Ghys and de la Harpe, "Infinite groups as geometric objects".

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