## Cubulating spaces with walls.(English)Zbl 1131.20030

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

### Citations:

Zbl 0916.51019; Zbl 0634.20015
Full Text:

### References:

 [1] H J Bandelt, J Hedlíková, Median algebras, Discrete Math. 45 (1983) 1 · Zbl 0506.06005 [2] I Chatterji, G Niblo, From wall spaces to $$\mathrm{CAT}(0)$$ cube complexes, Internat. J. Algebra Comput. 15 (2005) 875 · Zbl 1107.20027 [3] V Chepoi, Graphs of some $$\mathrm{CAT}(0)$$ complexes, Adv. in Appl. Math. 24 (2000) 125 · Zbl 1019.57001 [4] V Gerasimov, Fixed-point-free actions on cubings, Siberian Adv. Math. 8 (1998) 36 · Zbl 0912.20028 [5] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [6] F Haglund, F Paulin, Simplicité de groupes d’automorphismes d’espaces à courbure négative, Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 181 · Zbl 0916.51019 [7] G A Niblo, M A Roller, Groups acting on cubes and Kazhdan’s property (T), Proc. Amer. Math. Soc. 126 (1998) 693 · Zbl 0906.20024 [8] G Niblo, L Reeves, Groups acting on $$\mathrm{CAT}(0)$$ cube complexes, Geom. Topol. 1 (1997) · Zbl 0887.20016 [9] G A Niblo, L D Reeves, Coxeter groups act on $$\mathrm{CAT}(0)$$ cube complexes, J. Group Theory 6 (2003) 399 · Zbl 1068.20040 [10] M Roller, Poc-sets, median algebras and group actions: An extended study of Dunwoody’s construction and Sageev’s theorem, preprint (1998) [11] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. $$(3)$$ 71 (1995) 585 · Zbl 0861.20041 [12] D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150 · Zbl 1071.20038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.