Acylindrical actions on CAT(0) square complexes

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Abstract


For group actions on hyperbolic CAT(0) square complexes, we show that the acylindricity of the action is equivalent to a weaker form of acylindricity phrased purely in terms of stabilisers of points, which has the advantage of being much more tractable for actions on non-locally compact spaces. For group actions on general CAT(0) square complexes, we show that an analogous characterisation holds for the so-called WPD condition. As an application, we study the geometry of generalised Higman groups on at least 5 generators, the first historical examples of finitely presented infinite groups without non-trivial finite quotients. We show that these groups act acylindrically on the CAT (–1) polygonal complex naturally associated to their presentation. As a consequence, such groups satisfy a strong version of the Tits alternative and are residually F2-free, that is, every element of the group survives in a quotient that does not contain a non-abelian free subgroup.
Original languageEnglish
Pages (from-to)335–369
Number of pages35
JournalGroups, Geometry, and Dynamics
Volume15
Issue number1
Early online date25 Mar 2021
DOIs
Publication statusPublished - 2021

Keywords

  • Acylindrical actions
  • CAT.0/ cube complexes
  • Higman group
  • Tits alternative

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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