Acylindrical actions on CAT(0) square complexes

TL;DR

This paper characterizes acylindrical actions on CAT(0) square complexes, relates it to stabilizer-based conditions, and applies these results to analyze the geometry and algebraic properties of generalized Higman groups.

Contribution

It provides a new characterization of acylindricity and WPD conditions for actions on CAT(0) square complexes, and applies this to study generalized Higman groups.

Findings

Acylindricity is equivalent to a stabilizer-based condition in hyperbolic CAT(0) square complexes.

Generalized Higman groups act acylindrically on associated CAT(-1) complexes.

These groups satisfy a strong Tits alternative and are residually F_2-free.

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