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? 



Does every non-amenable group contain a 2-generated non-amenable subgroup?
The answer is "no". There are non-amenable Golod-Shafarevich groups where every 2-generated subgroup is finite. See my answer [here.](https://mathoverflow.net/questions/137678/non-amenable-groups-with-arbitrarily-large-tarski-number/137715#137715) 


