As far as I know, it is unknown whether Thompson's group F has Yu's Property A (that is, whether it is exact) or not. See for instance this MO question. The question is said to be open at several places in the litterature, but these references are several years old.
I would like to know if someone is aware of a recent development. Thanks for your help !
For the definition of Property A, see G. Yu, Invent. math. 139 (2000), 201-240.
It is unknown. I (and some others) believe it is as hard as amenability. There were two approaches to this problem. One was discussed in Arzhantseva, Guba, Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space. Using that approach one would need to construct an embedding of $F$ into a Hilbert space with compression function $\gg \sqrt{n}$. That approach was killed in Gournay's The Liouville property and Hilbertian compression (Numdam) where it was proved that the compression function cannot exceed $\sqrt{n}$. Another approach from Dranishnikov and Sapir's On the dimension growth of groups uses the so-called dimension growth. We hoped that the dimension growth of $F$ is subexponential which would imply $A$ (see Ozawa, Metric spaces with subexponential asymptotic dimension growth). But it turned out to be not quite the case, although see this Corrigendum to “On the dimension growth of groups” [J. Algebra 347 (1) (2011) 23–39], by Dranishnikov and Sapir. The idea of using various dimension growth functions is still alive though. See, for example, Dranishnikov and Zarichnyi, Asymptotic dimension, decomposition complexity, and Haver's property C
