# Examples of non-cubulated hyperbolic groups

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?

• A weaker notion, "PW" is just acting properly on a CAT(0) cube complex (possibly infinite-dimensional). It's much weaker than cubulable, still not satisfied by all hyperbolic groups (see AGenevois's answer), while satisfied by many groups. – YCor Dec 26 '19 at 11:31
• The Ollivier-Wise Rips machine produces hyperbolic groups with an infinite Property T subgroup and arbitrary f.p. quotient (hence possibly without Property T). These hyperbolic groups don't have Property PW, hence are not cubulable. – YCor Dec 28 '19 at 18:09

If a group $$G$$ satisfies Kazhdan's property (T), then any action of $$G$$ on a CAT(0) cube complex has a global fixed point. See Niblo and Roller's article Groups acting on cubes and Kazhdan's Property (T). Examples of hyperbolic groups which satisfy this property include:

• Uniform lattices in quaternionic hyperbolic spaces.
• Random groups in Gromov's model for some density.

Also, by a theorem due to Gromov, any hyperbolic group admits a quotient which is hyperbolic and has property (T).

• Do you know if uniform lattices in quaternionic hyperbolic spaces have torsion or are torsion free? – Yaniv Shakhar Dec 27 '19 at 11:22
• I am not quite familiar with lattices in quaternionic hyperbolic spaces, but there are reflections in such spaces, so I would say that there exist torsion-free lattices and lattices with torsion. – AGenevois Dec 27 '19 at 12:12
• @YanivShakhar: They're linear, and hence virtually torsion free. Probably there are explicit examples of quaterionionic hyperbolic lattices with torsion, but if you want to construct a hyperbolic group with torsion and property (T), just take any non-trivial element $g$ and kill a sufficiently high power $g^n$. – HJRW Dec 28 '19 at 16:46

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.

• That they're Haagerup is a 1974 result of Faraut-Harzallah. – YCor Dec 28 '19 at 17:52
• 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. – Misha Dec 28 '19 at 22:51
• @Misha — sure, but I wouldn’t class those as genuinely “different” examples. I agree that 2-dimensional examples would be particularly nice to have. – HJRW Dec 29 '19 at 6:20