On regular CAT(0) cube complexes and the simplicity of automorphism groups of rank-one CAT(0) cube complexes

Abstract

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 [20] and by Haglund and Paulin [15].