The simplicial boundary of a CAT(0) cube complex

Abstract

For a CAT(0) cube complex $X$, we define a simplicial flag complex ${\partial }_{△}X$, called the simplicial boundary, which is a natural setting for studying nonhyperbolic behavior of $X$. We compare ${\partial }_{△}X$ to the Roller, visual and Tits boundaries of $X$, give conditions under which the natural CAT(1) metric on ${\partial }_{△}X$ makes it isometric to the Tits boundary, and prove a more general statement relating the simplicial and Tits boundaries. The simplicial boundary ${\partial }_{△}X$ allows us to interpolate between studying geodesic rays in $X$ and the geometry of its contact graph $\mathrm{\Gamma }X$, 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 $X$ using ${\partial }_{△}X$. Finally, we rephrase the rank-rigidity theorem of Caprace and Sageev in terms of group actions on $\mathrm{\Gamma }X$ and ${\partial }_{△}X$ and state characterizations of cubulated groups with linear divergence in terms of $\mathrm{\Gamma }X$ and ${\partial }_{△}X$.