Product separability for special cube complexes

TL;DR

This paper proves that in groups acting virtually specially on CAT(0) cube complexes, products of convex-cocompact subgroups are separable, extending known results to more general settings and providing applications to group actions on contact graphs.

Contribution

It establishes the separability of products of convex-cocompact subgroups in a broad class of groups acting on CAT(0) cube complexes, generalizing previous partial results.

Findings

Products of convex-cocompact subgroups are separable in virtually special groups.

Extension of separability results beyond hyperbolic and cocompact cases.

Applications to actions on contact graphs and other graphs by cubulated groups.

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).