EFFECTIVE FLIPPING, SKEWERING AND RANK RIGIDITY FOR CUBULATED GROUPS WITH FACTOR SYSTEMS

Relying on work of Caprace and Sageev [Geom. Funct. Anal. 21 (2011), pp. 851–891], 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 compact special 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 [Amer. J. Math. 135 (2013), pp. 1087–1116] and provides an effective rank-rigidity dichotomy in the context of HHGs via Durham-Hagen-Sisto [Geom. Topol. 21 (2017), pp. 3659–3758]. 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 curtain recently introduced by Petyt, Spriano and the author in [Adv. Math. 450 (2024), p. 66] and Zalloum [Injectivity, cubical approximations and equivariant wall structures beyond cat(0) cube complexes, 2023]. 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.