Product separability for special cube complexes
Abstract
We prove that if a group admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of is separable. Previously, this was only known for products of three subgroups, or in the case where is hyperbolic, or in some other more technical cases with additional assumptions on the subgroups (plus these previous results assume that the action of 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).
Cite
@article{arxiv.2412.08248,
title = {Product separability for special cube complexes},
author = {Sam Shepherd},
journal= {arXiv preprint arXiv:2412.08248},
year = {2025}
}
Comments
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local-finiteness assumption; v3: The local-finiteness assumption has been removed from Theorem 1.1; v4: Theorem 1.3 has been replaced by Proposition 1.3 and Corollary 1.4; v5: Corollary 1.2 has been generalised.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