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Relying on work of Caprace and Sageev \cite{capracesageev:rank}, we providean effective form of rank rigidity in the context of groups virtually actingfreely cocompactly on a CAT(0) cube complex with a factor system. We accomplishthis by exhibiting a special pair of hyperplanes that can be skewered uniformlyquickly. Furthermore, for virtually special compact groups, we prove aneffective omnibus theorem and provide a trichotomy implying a strong form of aneffective Tits alternative. More generally, we provide a recipe for producingshort Morse elements generating free stable subgroups in any virtuallytorsion-free hierarchically hyperbolic group (HHG) which recovers Mangahas'work \cite{MangahasRecipie} and provides an effective rank-rigidity dichotomyin the context of HHGs via Durham-Hagen-Sisto \cite{Durham2017-ce}. Part of ouranalysis involves showing that Caprace-Sageev's cubical tools of flipping andskewering 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 Morseelement proceeds by exhibiting a special pair of curtains in the underlying HHGthat can be skewered uniformly quickly.

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