Why can't a non-amenable Torsion group have a non-abelian free subgroup

Question

I'm trying to work through some proofs in and article which is talking about Tarski numbers. There is a proposition which says that For a group G one has T(G)=4 if and only if G contains a non-abelian free subgroup. (Call this prop. a) I understand this. In the proof of the next proposition it says that by prop.a we know that The Tarski number of a non-amenable torsion group has Tarski number greater than or equal to 5. This implies that it's obvious that a non-amenable torsion group cannot have a non-abelian free subgroup. I may be missing something obvious but I simply don't know why this is. I'll be grateful for any answers, even if you can only give a push in the right direction. Thankyou :3 P.s. If you want to know the article it is arXiv:1603.04212v1 [math.GR] 14Mar2016 and I am looking at propositions 20 and 21.