Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

Abstract

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.