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](https://mathoverflow.net/questions/296246/). 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](https://doi.org/10.1007/s002229900032).



Does Thompson's group F have Yu's property A?
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](https://arxiv.org/abs/math/0411605)*. 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](https://doi.org/10.5802/aif.3067)* ([Numdam](http://www.numdam.org/item/AIF_2016__66_6_2435_0/)) 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](https://arxiv.org/abs/1008.3868)* 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](https://arxiv.org/abs/1108.3165))*. 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]](https://doi.org/10.1016/j.jalgebra.2012.07.048)*, 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](https://doi.org/10.1016/j.topol.2014.02.035)*


