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Dec 29, 2019 at 6:20 comment added HJRW @Misha — sure, but I wouldn’t class those as genuinely “different” examples. I agree that 2-dimensional examples would be particularly nice to have.
Dec 28, 2019 at 22:51 comment added Misha You can get more examples by taking free products of uniform complex-hyperbolic lattices in $PU(n,1)$, $n\ge 2$. What is unknown, I think, is the existence of 2-dimensional hyperbolic groups which have the Haagerup property but do not admit cubulations.
Dec 28, 2019 at 17:52 comment added YCor That they're Haagerup is a 1974 result of Faraut-Harzallah.
Dec 28, 2019 at 16:56 history answered HJRW CC BY-SA 4.0