In his 2008 paper *Hyperlinear and Sofic Groups: A Brief Guide*, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been answered in the meanwhile?


A short recap of the relevant definitions: A group is hyperlinear (sofic) if it embeds into the metric ultraproduct of unitary groups equipped with the normalized Hilbert-Schmidt distance (symmetric groups with the normalized Hamming distance).


A group has the Haagerup property if there is a sequence of positive definite functions that vanish at infinity and converge pointwise to the constant function $1$.



Are groups with the Haagerup property hyperlinear?
Thompson's group $F$ has the Haagerup property [1], but it is not known if it is hyperlinear according to Narutaka Ozawa's comment.


[1] Farley. *Finiteness and CAT(0) properties of diagram groups*. Topology, 2003.


