Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi\_1(M^4)$.


Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube complex?


Note that by work of Bergeron Wise based on work of Kahn-Markovic the answer to the question is always positive for $\Gamma=\pi\_1(M^3)$, where $M^3$ is any hyperbolic $3$-manifold. However, if instead $\Gamma$ is a co-compact lattice in $U(2,1)$, the answer is always negative (by Delzant Py (I guess)).



Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes
This question is certainly open in general. I don't know if anyone has formally expressed an 'expectation' in print, but you might be interested in the following pieces of positive evidence.


I believe that all known examples of such 4-manifolds essentially come either from arithmetic constructions or from Coxeter groups. Haglund--Wise showed that all hyperbolic Coxeter groups are cocompactly cubulated (indeed virtually special) [3], and Bergeron--Haglund--Wise showed that 'standard' cocompact arithmetic lattices in $SO(n,1)$ are virtually special [1]. So most known examples are certainly cocompactly cubulated.


One might worry that closed hyperbolic 3-manifolds are too rigid to be constructed in large numbers, in the way that Kahn--Markovic did with surfaces. But Calegari and I were able to construct very many acylindrical hyperbolic 3-manifolds in random groups [2]. (They don't cubulate for other reasons, but it makes the point that a single group may contain many rigid subgroups.) 


[1] Bergeron, Nicolas(F-PARIS6-IMJ); Haglund, Frédéric(F-PARIS11-M); Wise, Daniel T.(3-MGL)
Hyperplane sections in arithmetic hyperbolic manifolds. (English summary)
J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431–448.


[2] Calegari, D; Wilton, H.
3-manifolds everywhere, <https://arxiv.org/abs/1404.7043>


[3] Haglund, Frédéric(F-PARIS11-M); Wise, Daniel T.(3-MGL)
Coxeter groups are virtually special. (English summary)
*Adv. Math.* 224 (2010), no. 5, 1890–1903


