Groups acting on CAT(0) cube complexes

Abstract

We show that groups satisfying Kazhdan’s property $(T)$ have no unbounded actions on finite dimensional $CAT\left(0\right)$ 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.