Coarse decompositions of boundaries for CAT(0) groups

In this work we introduce a new combinatorial notion of boundary $Re C$ of an $omega$-dimensional cubing $C$. $Re C$ 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 -- $HH$ 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 $HH$ induces a function $rho:bd XtoRe C$. We develop a notion of uniformness for $HH$, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair $(X,HH)$ admits a geometric action by a group $G$, then the fibers of $rho$ form a stratification of $bd X$ graded by the order structure of $Re C$. 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 co-compactness of the action of $G$ on $C$ in terms of $rho$, generalizing a result of Williams, previously known only for Coxeter groups.