TITLE: Examples of non-cubulated hyperbolic groups
QUESTION [8 upvotes]: What is known regarding which hyperbolic groups are cubulated?
I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.
My impression is that not all of them are, but I didn't manage to find references with a counterexample.
Are there known ways to create non-cubulated hyperbolic groups? 
Are there famous examples of non-cubulated groups?

REPLY [6 votes]: As @AGenevois says in his answer, the standard examples of non-cubulated hyperbolic groups are those with Kazhdan's property (T), such as quaternionic hyperbolic lattices.
Complex hyperbolic lattices (in dimension >2) provide a more delicate class of examples.  On the one hand, they are not cubulable, by a theorem of Delzant--Py.  On the other hand, they are known to have the Haagerup property (I think this is proved in the book by Bekka--de la Harpe--Valette), so they also don't have property (T).
As far as I know, they are the only class of examples of hyperbolic groups known to be Haagerup but non-cubulable. I'd be interested to hear of others.