A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Some well-known examples are free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds. I will show in the talk that semidirect products of hyperbolic groups with $\mathbb{Z}$ which are again hyperbolic are cubulated, and give some consequences.
Two prominent examples of our setup are
Both these classes of groups are known to be cubulated by outstanding works. Our proof uses these two noteworthy results as building blocks and places them in a unified framework. Based on joint work with François Dahmani and Jean Pierre Mutanguha.