Just for curiosity I have done a quick web-search and I have seen that some people are studying manifolds with amenable fundamental group. On the other hand, any finitely presented group and then, in particular, Thompson's group arises as the fundamental group of a compact 4-manifold. So I have a series of questions:
Question 1: Is there any characterizing property of 4-manifolds having amenable fundamental group?
Question 2: Is there any explicit way to construct a 4-manifold whose fundamental group is the Thompson group $F$?
Mark Sapir's answer below says that something is known, but not enough to help for proving/disproving amenability of $F$.
The last question is very general:
Question 3: Is there anybody who's approaching the amenability of $F$ in this way, i.e. studying particular topological properties of a 4-manifold whose fundamental group is $F$?
Also in this case Mark Sapir's answer below shows that some (unsuccessful) attempt was made. Does anybody know something more?
Thanks in advance,