Property A and CAT(0) cube complexes

Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finitedimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable.