  We provide a necessary and sufficient condition on a finite flag simplicial
complex, L, for which there exists a unique CAT(0) cube complex whose vertex
links are all isomorphic to L. We then find new examples of such CAT(0) cube
complexes and prove that their automorphism groups are virtually simple. The
latter uses a result, which we prove in the appendix, about the simplicity of
certain subgroups of the automorphism group of a rank-one CAT(0) cube complex.
This result generalizes previous results by Tits and by Haglund and Paulin.
