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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 -free, that is, every element of the group survives in a quotient that does not contain a non-abelian free subgroup.
Cite this article
Alexandre Martin, Acylindrical actions on CAT(0) square complexes. Groups Geom. Dyn. 15 (2021), no. 1, pp. 335–369DOI 10.4171/GGD/600