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  • 7 - A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes

  • from Part Three - Research Articles
  • Edited by Mark Hagen, University of Bristol, Richard Webb, University of Cambridge, Henry Wilton, University of Cambridge
  • Book:
    Beyond Hyperbolicity
    Published online:
    22 June 2019
    Print publication:
    11 July 2019, pp 160-178

  • Summary

    We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a group acting essentially and non-elementarily on a finite dimensional irreducible CAT(0) cube complex is acylindrically hyperbolic if there exist two hyperplanes whose stabilisers intersect along a finite subgroup. We also give further conditions on the geometry of the complex so that the result holds if we only require the existence of a single pair of points whose stabilisers intersect along a finite subgroup.