Effective flipping, skewering and rank rigidity for cubulated groups with factor systems

Abstract

Relying on work of Caprace and Sageev \cite{capracesageev:rank}, we provide an effective form of rank rigidity in the context of groups virtually acting freely cocompactly on a CAT(0) cube complex with a factor system. We accomplish this by exhibiting a special pair of hyperplanes that can be skewered uniformly quickly. Furthermore, for virtually special compact groups, we prove an effective omnibus theorem and provide a trichotomy implying a strong form of an effective Tits alternative. More generally, we provide a recipe for producing short Morse elements generating free stable subgroups in any virtually torsion-free hierarchically hyperbolic group (HHG) which recovers Mangahas' work \cite{MangahasRecipie} and provides an effective rank-rigidity dichotomy in the context of HHGs via Durham-Hagen-Sisto \cite{Durham2017-ce}. Part of our analysis involves showing that Caprace-Sageev's cubical tools of flipping and skewering can be applied to any HHG using the notion of a \emph{curtain} recently introduced by Petyt, Spriano and the author in \cite{PSZCAT}, \cite{Zalloum23_injectivity}. Indeed, our route to producing a short Morse element proceeds by exhibiting a special pair of curtains in the underlying HHG that can be skewered uniformly quickly.