## Geometry & Topology

### Groups acting on CAT(0) cube complexes

#### Abstract

We show that groups satisfying Kazhdan’s property $(T)$ have no unbounded actions on finite dimensional $CAT(0)$ cube complexes, and deduce that there is a locally $CAT(−1)$ Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally $CAT(0)$ cube complex.

#### Article information

Source
Geom. Topol., Volume 1, Number 1 (1997), 1-7.

Dates
Accepted: 6 February 1997
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882841

Digital Object Identifier
doi:10.2140/gt.1997.1.1

Mathematical Reviews number (MathSciNet)
MR1432323

Zentralblatt MATH identifier
0887.20016

#### Citation

Niblo, Graham A; Reeves, Lawrence. Groups acting on CAT(0) cube complexes. Geom. Topol. 1 (1997), no. 1, 1--7. doi:10.2140/gt.1997.1.1. https://projecteuclid.org/euclid.gt/1513882841

#### References

• M Bridson, Geodesics and curvature in metric simplicial complexes, from: “Group Theory from a Geometrical Viewpoint”, E Ghys et al (eds.), World Scientific (1991) 373–463
• M Bridson, A Haefliger, Metric spaces of non-positive curvature, in preparation
• M Bozejko, T Janusckiewicz, R T Spatzier, Infinite Coxeter groups do not have Kazhdan's property, J. Operator Theory 19 (1988) 63–37
• M Davis, Buildings are CAT($0$), from: “Geometric methods in group theory”, P H Kropholler, G A Niblo and R Stohr (eds.), LMS Lecture Note Series, Cambridge University Press
• M Gromov, Hyperbolic groups, from: “Essays in group theory”, S M Gersten (ed.), MSRI Publ. 8, Springer–Verlag (1987) 75–267
• P de la Harpe, A Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Asterisque 175 (1989), Société Mathématique de France
• G A Niblo, L D Reeves, Coxeter groups act on CAT($0$) cube complexes, preprint
• M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Maths. Soc. (3) 71 (1995) 585–617