# Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1$ weak$^*$ in $B_2(\Gamma)$ i.e $w(\phi_i) \to w(1)$ for every $w \in Q(\Gamma)$.

A wanted to know, if anything is known in the literature about the Thompson's group $V$, having property-AP?

It would be great, if I am directed to some research papers regarding the same.

Thanks for the help!!

• AP passes to (closed) subgroups, so if V has AP then so does F. The AP implies exactness, so if F has AP then it is exact. I believe it is still an open question if F is exact (see Ozawa's comment to this MO question mathoverflow.net/questions/202835 ) and from this I would deduce: there is currently no proof that V has AP. Mar 26, 2018 at 23:02
• Conversely, the list of "explicit" groups that fail the AP seems to be quite restricted -- this is now known for lattices in higher rank Lie groups, but this requires technques from harmonic analysis that do not seem to be applicable to F, T or V. So I would guess that no one has yet proved V fails to have AP! Mar 26, 2018 at 23:08
• @YemonChoi, you are correct.
– user6976
Dec 1, 2019 at 2:10