# Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

### Jason Behrstock

Lehman College, CUNY, Bronx, USA### Mark F. Hagen

University of Cambridge, UK

## 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 $∂_{△}X$ 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 and 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 $∂_{△}X$. 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, $∂_{△}X$. Finally, we construct, for each $n≥0,k≥2$, 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.

## Cite this article

Jason Behrstock, Mark F. Hagen, Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. Groups Geom. Dyn. 10 (2016), no. 2, pp. 649–707

DOI 10.4171/GGD/360