Geometry & Topology, Vol. 1 (1997) Paper no. 1, pages 1-7.

Groups acting on CAT(0) cube complexes

Graham Niblo and Lawrence Reeves

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.

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

AMS subject classification. Primary: 20F32 Secondary: 20E42, 20G20

DOI: 10.2140/gt.1997.1.1

E-print: arXiv:math.GR/9702231

Submitted to GT on October 28, 1996. Final version accepted on February 6, 1997.

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Graham Niblo
Faculty of Mathematical Studies
University of Southampton
Southampton, SO17 1BJ, UK

Lawrence Reeves
Institute of Mathematics
Hebrew University of Jerusalem
Givat Ram
Jerusalem 91904, Israel

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