Isometry groups of CAT(0) cube complexes

Original · EN

Given a CAT(0) cube complex X, we show that if Aut(X) ≠ Isom(X) then there exists a full subcomplex of X which decomposes as a product with Rⁿ. As applications, we prove that if X is δ-hyperbolic, cocompact and 1-ended, then Aut(X) = Isom(X) unless X is quasi-isometric to H², and extend the rank-rigidity result of Caprace-Sageev to any lattice Γ≤ Isom(X).