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It is known that there are non-amenable groups not containing $F_2$, the free group on two generators; for example, Olshanskii's group. But does every non-amenable group contain a 2-generated non-amenable subgroup?

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The answer is "no". There are non-amenable Golod-Shafarevich groups where every 2-generated subgroup is finite. See my answer here.

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