Coarse decompositions for boundaries of CAT(0) groups

In this work we introduce a new combinatorial notion of boundary <C of an ω-dimensional cubing C. <C 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 DavisMoussong complex of a finite rank Coxeter group), we show how H induces a function ρ : ∂∞X → <C. 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 <C. 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.