Given a finite CAT(0) cubical complex, we define flag simplicial complex associated to it, called the crossing complex. We show that holds much of combinatorial information original complex: for example, hyperplanes in correspond vertex links and is balanced if only cubically balanced. The most significant result sets f-vectors complexes are equal, up an invertible linear transformation.

We prove that a minimal disc in CAT(0) space is local embedding away from finite set of “branch points”. On the way we establish several basic properties surfaces: monotonicity area densities, density bounds, limit theorems and existence tangent maps. As an application, Fáry–Milnor’s theorem setting.

We show that groups satisfying Kazhdan’s property (T) have no unbounded actions on nite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(−1) Riemannian manifold which is not homotopy equivalent to any nite dimensional, locally CAT(0) cube complex. AMS Classi cation numbers Primary: 20F32 Secondary: 20E42, 20G20