The simplicial boundary of a CAT(0) cube complex

Hagen, Mark F¹

¹Department of Mathematics, University of Michigan, 530 Church Street, 1859 East Hall, Ann Arbor, MI 48109, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 3, pp. 1299-1367

Résumé

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

Détail

DOI : 10.2140/agt.2013.13.1299

Classification : 05C25, 20F65, 57M99
Keywords: CAT(0) cube complex, contact graph, divergence, rank-one isometry, simplicial boundary

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