I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with boundary embedded in $\mathbb{R}^3$, but would be happy to hear any answers to this question. If I had to name one concrete question, it would be:
Question: Can you cubulate every compact 3-manifold $M \subset \mathbb{R}^3$? If not, which ones can you cubulate? What are some specific examples of non-cubulable $M$?
I am aware of some scattered results, e.g.,
- all hyperbolic 3-manifolds are cubulable (discussed in Sections 4.5, 4.6 of this paper)
- there is some discussion about cubulability of Kähler groups/Kähler manifolds here
- there is a characterization in terms of the boundary here
Apart from the result about hyperbolic 3-manifolds, I find it quite hard to connect largely algebraic results like these back to a more concrete geometric/topological picture.