Acylindrical actions on CAT(0) square complexes

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 rst historical examples of nitely presented in nite groups without non-trivial nite 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. Acylindrical actions were rst considered by Sela for groups acting on simplicial trees [27]. In Sela's terminology, given a (minimal) action of a group on a simplicial tree, the action is said to be acylindrical if there exists an integer k ≥ 1 such that no non-trivial element of the group xes pointwise two points at distance at least k. This de nition was extended to actions on arbitrary geodesic metric spaces by Bowditch [4], in his study of the action of the mapping class group of a closed hyperbolic surface on its associated curve complex. Recall that an action of a group G on a metric space X is acylindrical if for every r ≥ 0 there exist constants L(r), N(r) ≥ 0 such that for every points x, y of X at distance at least L(r), there are at most N(r) elements h of G such that d(x, hx), d(y, hy) ≤ r. For r = 0, one recovers Sela's de nition of acylindricity, at least for torsion-free groups. As noticed by Bowditch [4], in the case of group actions on simplicial trees, acylindricity is equivalent to this weaker acylindricity condition at r = 0. Groups that act acylindrically on a hyperbolic spaces share many features with relatively hyperbolic groups, and techniques from dynamics in negative curvature are available to study them. Unfortunately, acylindrical actions on hyperbolic spaces seldom appear naturally in geometric group theory. However, weaker but more frequent forms of acylindricity are su cient to apply this circle of ideas, as noticed by various authors. In a nutshell, the 2010 Mathematics subject classi cation. 20F65