Is the property of not containing the free group on two generators invariant under quasiisometry? Amenability is, so if there is a counterexample it is also a solution to the von NeumannDay problem (which of course already has a solution).
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It is a famous open problem. Akhmedov in MR2424177 claimed he could prove that the answer is "no". No proof exists, so I guess he discovered a gap in his argument. 

