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 ${\partial }_{\Delta }\mathbf{X}$.

Lemma 3.18 was not stated correctly. This is fixed, and a minor adjustment to the beginning of the proof of Theorem 3.19 has been made as a result. Statements other than 3.18 do not need to change. I thank Abdul Zalloum for the correction. See also: arXiv:2004.01182 (this version differs from previous only by addition of the preceding link, at administrators' request)

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• Boundaries and automorphisms of hierarchically hyperbolic spaces
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• On hierarchical hyperbolicity of cubical groups
• Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups
• Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
• Divergence of CAT(0) Cube Complexes and Coxeter Groups
• Roller boundaries for median spaces and algebras
• The Tits alternative for finite rank median spaces
• The geometry of the curve graph of a right-angled Artin group
• Large facing tuples and a strengthened sector lemma
• Superrigidity of actions on finite rank median spaces
• Cocompactly cubulated crystallographic groups
• Malnormality and join-free subgroups in right-angled Coxeter groups
• Divergence of finitely presented subgroups of CAT(0) groups
• Corrigendum to "The simplicial boundary of a CAT(0) cube complex''
• Homotopy equivalent boundaries of cube complexes
• Infinite reduced words and the Tits boundary of a Coxeter group
• Acylindrical hyperbolicity of cubical small-cancellation groups
• Proper actions of Grigorchuk groups on a CAT(0) cube complex
• Divergence, thickness and hypergraph index for general Coxeter groups