Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 2 / c / Solution

The necessary and sufficient condition is the transverse-wall cocompactness criterion: there must be finitely many $G$-orbits of finite transverse wall collections. Equivalently, their cardinalities must be uniformly bounded and, for every cardinality, there must be only finitely many orbits.

Indeed, an $n$-cube of the dual cube complex of a wallspace is dual to an $n$-element collection of pairwise crossing walls, and this correspondence respects the $G$-action and passage to faces. If there are finitely many orbits of transverse collections, there are finitely many cube orbits, so the quotient $G\backslash C$ is a finite cube complex and is compact. Conversely, if the action is cocompact, a compact fundamental set meets only finitely many open unit cubes. Hence there are finitely many cube orbits and therefore finitely many orbits of their dual transverse wall collections.