The simplicial boundary of a CAT(0) cube complex

Abstract

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_\Delta\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_\Delta\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_\Delta\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_\Delta\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $\Gamma\mathbf X$, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $\mathbf X$ using $\partial_\Delta\mathbf X$. Finally, we rephrase the rank-rigidity theorem of Caprace-Sageev in terms of group actions on $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$ and state characterizations of cubulated groups with linear divergence in terms of $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$.