Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

TL;DR

This paper characterizes relatively hyperbolic groups acting on CAT(0) cube complexes through their simplicial boundaries, relates boundary cut-points to asymptotic cone properties, and constructs examples of groups with specific thickness and divergence properties.

Contribution

It provides a new boundary-based criterion for relative hyperbolicity in cubical groups and links boundary topology to algebraic thickness, also constructing diverse examples with controlled geometric properties.

Findings

Relatively hyperbolic groups correspond to boundary decompositions.

Cut-points in asymptotic cones relate to boundary skeletons.

Constructed groups with specified thickness and divergence.

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.