Cubulability of closed hyperbolic 4-manifolds

Determine whether every closed hyperbolic 4-manifold is cubulable; specifically, establish whether the fundamental group of any closed hyperbolic 4-manifold admits a geometric (proper and cocompact) action by isometries on a CAT(0) cube complex.

Background

This work shows that for torsion-free uniform lattices acting on H^n, with n ≤ 3 or for arithmetic lattices of simplest type, the lattice action can be metrically approximated by geometric actions on CAT(0) cube complexes, solving a conjecture of Futer and Wise in those cases. It also develops tools via co-geodesic currents to paper cubulations and their limits.

In contrast, the general cubulability of closed hyperbolic 4-manifolds (i.e., the existence of geometric actions of their fundamental groups on CAT(0) cube complexes) remains unsettled. The authors single out this case explicitly as open, highlighting a gap between the known results in low dimensions and arithmetic settings and the broader class of closed hyperbolic 4-manifolds.