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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 $1$.

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    $\begingroup$ I believe this is still open. $\endgroup$
    – Jon Bannon
    Jul 1, 2020 at 12:07
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    $\begingroup$ For an irreducible lattice in a nonlinear Lie group such as a finite cover of $\mathrm{SU}(2,1)^k$ I guess hyperlinearity is unknown. (I'm not sure if non-residually finite lattices are known in such groups, but it sounds plausible). $\endgroup$
    – YCor
    Jul 1, 2020 at 12:36
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    $\begingroup$ It's not known if Thompson's group F is hyperlinear. $\endgroup$
    – Narutaka OZAWA
    Jul 2, 2020 at 3:27
  • $\begingroup$ Thank you all for your comments. Especially @NarutakaOZAWA 's comment seems as good an answer I can expect. If you want to post it as an answer, I would be glad to accept it. $\endgroup$
    – MaoWao
    Jul 3, 2020 at 6:38

1 Answer 1

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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.

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