Coarse decompositions of boundaries for CAT(0) groups
| dc.creator | Guralnik, Dan | |
| dc.date | 2006-10-31 | |
| dc.date | 2007-12-02 | |
| dc.date.accessioned | 2026-07-07T08:46:16Z | |
| dc.date.available | 2026-07-07T08:46:16Z | |
| dc.description | In this work we introduce a new combinatorial notion of boundary $\Re C$ of an $ω$-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 $ρ:\bd X\to\Re C$. 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 $ρ$ 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 $ρ$, generalizing a result of Williams, previously known only for Coxeter groups. | |
| dc.description | 54 pages, 4 figures. Improved exposition, significantly strengthened results | |
| dc.identifier | https://arxiv.org/abs/math/0611006 | |
| dc.identifier | http://arxiv.org/abs/math/0611006 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143196 | |
| dc.subject | Group Theory | |
| dc.subject | 20F65, 20F67 (Primary), 20F55 (Secondary) | |
| dc.title | Coarse decompositions of boundaries for CAT(0) groups | |
| dc.type | text |