The simplicial boundary of a CAT(0) cube complex

TL;DR

This paper introduces the simplicial boundary of a CAT(0) cube complex, providing a new framework to analyze its non-hyperbolic behavior and relate it to existing boundaries and the contact graph.

Contribution

It defines the simplicial boundary, compares it with other boundaries, and explores its applications to divergence, contact graph properties, and rank-rigidity in cubulated groups.

Findings

The simplicial boundary is a natural setting for studying non-hyperbolic behavior.

Conditions are given for when the simplicial boundary is (quasi)isometric to the Tits boundary.

Characterizations of cubulated groups with linear divergence are provided.

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 $\partial_\Delta\mathbf X$.