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Groups, Geometry, and Dynamics

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Volume 15, Issue 1, 2021, pp. 335–369
DOI: 10.4171/GGD/600

Published online: 2021-03-25

Acylindrical actions on CAT(0) square complexes

Alexandre Martin[1]

(1) Heriot-Watt University, Edinburgh, UK

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.

Keywords: CAT(0) cube complexes, acylindrical actions, Higman group, Tits alternative

Martin Alexandre: Acylindrical actions on CAT(0) square complexes. Groups Geom. Dyn. 15 (2021), 335-369. doi: 10.4171/GGD/600