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,


Valerio



Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group
Thompson group has a Wirtinger-like presentation $x\_i^{x\_j}=x\_{i+1}, 0\le i < j\le 4$, a $C$-group in the terminology of Kuzʹmin, Yu. V. Groups of knotted compact surfaces, and central extensions. Mat. Sb. 187 (1996), no. 2, 81--102; translation in Sb. Math. 187 (1996), no. 2, 237–257 . Hence by a result of Kuzmin (also Gilbert, Howie and others) it is the fundamental group of the complement of a knotted $S^2$ inside $S^4$. This topological property of $F$ does not help in proving/disproving amenability because many amenable groups are $C$-groups as well (by other papers of Kuzmin). Of course some other properties of $F$ give some more information about compact manifolds with $F$ as fundamental group. For example $F$ has quadratic Dehn function by Guba, V. S. The Dehn function of Richard Thompson's group $F$ is quadratic. Invent. Math. 163 (2006), no. 2, 313–342. But this also does not help much for the amenability question. 


