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 ${\partial }_{△}\mathbf{X}$ 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 ${\partial }_{△}\mathbf{X}$. 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, ${\partial }_{△}\mathbf{X}$. Finally, we construct, for each $n\ge 0,k\ge 2$, 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.