The simplicial boundary of a CAT(0) cube complex

Abstract

For a CAT(0) cube complex X, we define a simplicial flag complex @ 4 X, called the simplicial boundary, which is a natural setting for studying nonhyperbolic behavior of X.We compare @ 4 X to the Roller, visual and Tits boundaries of X, give conditions under which the natural CAT(1) metric on @ 4 X makes it isometric to the Tits boundary, and prove a more general statement relating the simplicial and Tits boundaries.The simplicial boundary @ 4 X allows us to interpolate between studying geodesic rays in X and the geometry of its contact graph 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 @ 4 X.Finally, we rephrase the rank-rigidity theorem of Caprace and Sageev in terms of group actions on X and @ 4 X and state characterizations of cubulated groups with linear divergence in terms of X and @ 4 X.

Locations

• Algebraic & Geometric Topology - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• DataCite API - View