Geometry & Topology

Groups acting on CAT(0) cube complexes

Graham A Niblo and Lawrence Reeves

Full-text: Open access


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

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

Received: 28 October 1996
Accepted: 6 February 1997
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F32
Secondary: 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 20G20: Linear algebraic groups over the reals, the complexes, the quaternions

Kazhdan's property (T) Tits' buildings hyperbolic geometry CAT(0) cube complexes locally CAT(-1) spaces $Sp(n,1)$–manifolds


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.

Export citation


  • 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