Coarse decompositions of boundaries for CAT(0) groups, Electronic preprint arxiv:math/0611006

In this work we introduce a new combinatorial notion of boundary ℜC of an ω-dimensional cubing C. ℜC is defined to be the set of almostequality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation H of a CAT(0) space X (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how H induces a function ρ: ∂∞X → ℜC. We develop a notion of uniformness for H, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair (X, H) admits a geometric action by a group G, then the fibers of ρ form a stratification of ∂∞X modeled on the order structure of ℜC, and demonstrate connections between this structure and rectifiable paths on the Tits boundary of X. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of C, we give a condition for the co-compactness of the action of G on C in terms of ρ, generalizing of a result of Williams previously known only for Coxeter groups.

Coarse decompositions of boundaries for CAT(0) groups, Electronic preprint arxiv:math/0611006

In this work we introduce a new combinatorial notion of boundary ℜC of an ω-dimensional cubing C. ℜC is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation – or discrete system of halfspaces – H of a CAT(0) space X (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how H induces a function ρ: ∂∞X → ℜC. We develop a notion of uniformness for H, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair (X, H) admits a geometric action by a group G, then the fibers of ρ form a stratification of ∂∞X graded by the order structure of ℜC. We also show how this structure computes the components of the Tits boundary of X. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of C, we give a condition for the cocompactness of the action of G on C in terms of ρ, generalizing a result of Williams, previously known only for Coxeter groups.

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 ↦ → X � 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.

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 ↦ → X � 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.