Local finiteness of cubulations and CAT(0) groups

Let X be a geodesic space and G a group acting geometrically on X. A discrete halfspace system 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 such a system H, one uses the Sageev-Roller construction to form a cubing C(H). When H is invariant under G we have: Theorem A. X has a G-equivariant quasi-isometric embedding into C(H). The basic questions about C(H) are: when are all cubes in C(H) finite-dimensional? when is C(H) finite dimensional? when is it proper? when is C(H) G-co-compact (and hence G is biautomatic, by a result of Niblo and Reeves)? These questions were answered by Niblo-Reeves, Williams and Caprace for the case of Coxeter groups (W, R) acting on their Davis-Moussong complexes, with elements of H being the halfspaces defined by reflections. A significant role was played by the ‘parallel walls property’ of Coxeter groups, conjectured by Davis and Shapiro and proved by Brink and Howlett. It thus becomes natural to ask these questions whenever X is a CAT(0) space carrying a geometric action by a group G. In this paper we show that, when H has bounded chambers, the parallel walls property is equivalent to a condition we call uniformness, regarding the quality of approximation of boundary points by walls of H. Uniformness, as opposed to the parallel walls property, involves no explicit bounds. We prove: Theorem B. Let G be a group acting geometrically on a geodesic space X and suppose H is a discrete G-invariant halfspace system in X. If H is uniform, then C(H) is proper (locally-finite). In particular, C(H) does not contain infinite-dimensional cubes.