# Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group

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?

Valerio

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Since every group is the fundamental group of a $4$-manifold but most groups are not amenable, I can't see how this would buy you anything. One would need a $4$-manifold with some kind of extra structure that not all $4$-manifolds have, and I don't see how to do that with Thompson's group. – Andy Putman Oct 27 '11 at 18:49
Indeed my question contains many (maybe trivial) subquestions: 1) Does a 4-manifold with amenable fundamental group have any characterizing properties? 2) Is there an explicit way to construct a 4-manifold whose fundamental group is the Thompson group? – Valerio Capraro Oct 27 '11 at 19:18
@Valerio: it usually works better if you ask the subquestions explicitly :) – Mariano Suárez-Alvarez Oct 27 '11 at 19:31
About 2), the construction of Kuzmin is explicit. About 3): there were no attempts (successful or not) because the idea does not seem to be fruitful: there seems to be no info you can get from the manifold that you cannot obtain just by looking at the very simple defining relations of $F$. One can get more information viewing $F$ as a fundamental group of an infinite dimensional cubical complex with CAT(0) universal cover. – Mark Sapir Oct 27 '11 at 19:59
a-T-menability is a strong negation of property (T). The class of a-T-memable groups contains all amenable groups. Another approximation of amenability is G.Yu's property A. It is not known whether $F$ satisfies property A. That question seems to be as hard as amenability. Free groups and even all hyperbolic groups satisfy A. – Mark Sapir Oct 27 '11 at 21:35

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.