# Groups, Geometry, and Dynamics

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**Volume 10, Issue 2, 2016, pp. 649–707**

**DOI: 10.4171/GGD/360**

Published online: 2016-06-09

Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

Jason Behrstock^{[1]}and Mark F. Hagen

^{[2]}(1) Lehman College, CUNY, Bronx, USA

(2) University of Cambridge, UK

Let $G$ be a group acting geometrically on a CAT(0) cube complex $\mathbf X$. We prove first that $G$ is hyperbolic relative to the collection $\mathbb P$ of subgroups if and only if the simplicial boundary $\partial_{\triangle} \mathbf X$ 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 $\mathbb P$. 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 $\mathbf X$ to boundedness of the 1-skeleton of $\partial_{\triangle} \mathbf X$. We deduce characterizations of thickness and strong algebraic thickness of a group $G$ acting properly and cocompactly on the CAT(0) cube complex $\mathbf X$ in terms of the structure of, and nature of the $G$-action on, $\partial_{\triangle} \mathbf X$. Finally, we construct, for each $n\geq 0, k\geq 2$, 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.

*Keywords: *Thick metric space, thick group, cubulated group, cube complex, relatively hyperbolic group, boundary, simplicial boundary

Behrstock Jason, Hagen Mark: Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. *Groups Geom. Dyn.* 10 (2016), 649-707. doi: 10.4171/GGD/360