Local finiteness of cubulations for CAT(0) groups

Let X be a proper CAT(0) space. A halfspace system (or cubulation) of X is a set H of open halfspaces closed under h 7→ X r h and such that every x ∈ X has a neighbourhood intersecting only finitely many walls of H. Given a cubulation H, one uses the Sageev-Roller construction to form a cubing C(H). One setting in which cubulations were studied is that of a Coxeter group (W,R) acting on its Davis-Moussong complex, with elements of H being the halfspaces defined by reflections. For this setting, Niblo and Reeves had shown that C(H) is a finite-dimensional, locally-finite cubing. Their proof explicitly uses the ‘parallel walls property’ of Coxeter groups, proved by Brink and Howlett, and heavily relies on meticulous calculations with the root system associated with (W, R). We offer a generalization of their local finiteness result using the visual boundary of X, endowed with the cone topology. We introduce an asymptotic condition on H that we call uniformness, which is implied by the parallel walls property together with boundedness of chambers. In a sense, uniformness regards the way in which boundary points are approximated by the walls of H. We prove: Theorem A. Let G be a group acting geometrically on a CAT(0) space X and suppose H is a cubulation of X invariant under G and having no infinite transverse subset. If H is uniform, then C(H) is locally-finite.