Mathematics – Group Theory
Scientific paper
2006-10-30
Mathematics
Group Theory
25 pages, 3 figures. Improved exposition, significantly strengthened results (assumption on $\omega$-dimensionality may be dro
Scientific paper
Let X be a proper CAT(0) space. A halfspace system (or cubulation) of X is a set H of open halfspaces closed under closure-complementation and such that every point in 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 generalize their local finiteness result using the visual boundary of X, endowed with the cone topology. We introduce an asymptotic condition on H (`uniformness'), and show it is equivalent to H having the parallel walls property together with boundedness of chambers. Uniformness regards the way in which boundary points are approximated by the walls of H. We prove that if G is a group acting geometrically on a CAT(0) space X and H is a uniform cubulation of X invariant under G, then C(H) is locally-finite. We also show that the obvious map of $X$ into $C(H)$ is a quasi-isometry, and use this to give quantitative results strengthening the parallel walls property.
No associations
LandOfFree
Local finiteness for cubulations of CAT(0) groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Local finiteness for cubulations of CAT(0) groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Local finiteness for cubulations of CAT(0) groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-47411