Product separability for special cube complexes

Abstract

We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case where $G$ is hyperbolic, or in some other more technical cases with additional assumptions on the subgroups (plus these previous results assume that the action of $G$ is cocompact). We also provide an application to the action of a virtually special cubulated group on its contact graph (and to some other actions of cubulated groups on graphs).

40 pages, 4 figures; v2: Theorem 1.1 has been generalised by replacing the cocompactness assumption with a local-finiteness assumption; v3: The local-finiteness assumption has been removed from Theorem 1.1; v4: Theorem 1.3 has been replaced by Proposition 1.3 and Corollary 1.4; v5: Corollary 1.2 has been generalised. To appear in Commentarii Mathematici Helvetici