A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes

TL;DR

This paper provides criteria based on stabilisers for determining when groups acting on finite-dimensional CAT(0) cube complexes are acylindrically hyperbolic, enhancing understanding of their geometric group properties.

Contribution

It introduces simple stabiliser-based criteria for acylindrical hyperbolicity in groups acting on CAT(0) cube complexes, including conditions involving hyperplanes and points.

Findings

Groups acting essentially and non-elementarily are acylindrically hyperbolic under certain stabiliser conditions.

Existence of hyperplanes with finite stabiliser intersections implies acylindrical hyperbolicity.

Additional geometric conditions allow similar conclusions with fewer assumptions.

Abstract

We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a group acting essentially and non-elementarily on a finite dimensional irreducible CAT(0) cube complex is acylindrically hyperbolic if there exist two hyperplanes whose stabilisers intersect along a finite subgroup. We also give further conditions on the geometry of the complex so that the result holds if we only require the existence of a single pair of points whose stabilisers intersect along a finite subgroup.