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 $F_2$-free, that is, every element of the group survives in a quotient that does not contain a non-abelian free subgroup.

中文翻译：

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CAT（0）方络合物上的圆柱作用

对于双曲CAT（0）方络合物上的群作用，我们表明该作用的acylindricity等效于仅用点的稳定器表示的较弱形式的acylindricity，其优点是，对于非on的作用更易处理-局部紧凑的空间。对于一般CAT（0）平方复合体上的组动作，我们证明了对所谓的WPD条件的相似刻画。作为应用，我们研究了至少5个生成器上的广义Higman群的几何，这是没有非平凡商的有限表示无限群的第一个历史示例。我们表明，这些组在与它们的表示自然相关的CAT（–1）多边形复合体上呈圆柱状作用。作为结果，