Cubulating spaces with walls. (English) Zbl 1131.20030

Nica, Bogdan

Algebr. Geom. Topol. 4, 297-309 (2004).

From the introduction: The elegant notion of a space with walls was introduced by F. Haglund and F. Paulin [Geom. Topol. Monogr. 1, 181-248 (1998; Zbl 0916.51019)]. Prototypical examples of spaces with walls are CAT(0) cube complexes, introduced by M. Gromov [in Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. The purpose of this note is to observe that every space with walls has a canonical embedding in a CAT(0) cube complex and, consequently, a group action on a space with walls extends naturally to a group action on a CAT(0) cube complex. The usefulness of this result is that spaces with walls are often easily identifiable by geometric reasons.

MSC:

20F65	Geometric group theory
53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20E42	Groups with a \(BN\)-pair; buildings

Keywords:

spaces with walls; median graphs; CAT(0) cube complexes; group actions

Citations:

Zbl 0916.51019; Zbl 0634.20015

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