#### Volume 1, issue 1 (1997)

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Groups acting on CAT(0) cube complexes

### Graham A Niblo and Lawrence Reeves

Geometry & Topology 1 (1997) 1–7
 arXiv: math.GR/9702231
##### Abstract

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

##### Keywords
Kazhdan's property (T), Tits' buildings, hyperbolic geometry, CAT(0) cube complexes, locally CAT(-1) spaces, $Sp(n,1)$–manifolds
##### Mathematical Subject Classification
Primary: 20F32
Secondary: 20E42, 20G20