Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

Abstract

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of P. As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cube complex X to boundedness of the 1-skeleton of the boundary of X. We deduce characterizations of thickness and strong algebraic thickness of a group G acting properly and cocompactly on the CAT(0) cube complex X in terms of the structure of, and nature of the G-action on, the boundary of X. Finally, we construct, for each n,k, infinitely many quasi-isometry types of group G such that G is strongly algebraically thick of order n, has polynomial divergence of order n+1, and acts properly and cocompactly on a k-dimensional CAT(0) cube complex.

Corrections according to referee report. Fixed proof of Theorem 4.3. To appear in "Groups, Geometry, and Dynamics"

References in corpus (1)

• Divergence in right-angled Coxeter groups

Cited by in corpus (11)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• On hierarchical hyperbolicity of cubical groups
• Top dimensional quasiflats in $CAT(0)$ cube complexes
• Thickness, relative hyperbolicity, and randomness in Coxeter groups
• Global Structural Properties of Random Graphs
• Divergence of CAT(0) Cube Complexes and Coxeter Groups
• Corrigendum to "The simplicial boundary of a CAT(0) cube complex''
• Homotopy equivalent boundaries of cube complexes
• Subdivision rules for special cubulated groups
• Divergence in right-angled Coxeter groups