Mathematics – Group Theory
Scientific paper
2012-01-04
Mathematics
Group Theory
Scientific paper
For a CAT(0) cube complex $\mathbf X$, we define a simplicial complex $\simp\mathbf X$, called the \emph{simplicial boundary}. $\simp\mathbf X$ differs from the Tits boundary, the Roller boundary, and the visual boundary of $\mathbf X$. The simplicial boundary is the natural setting for studying non-hyperbolic behavior of $\mathbf X$. The simplicial boundary allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph}, which is known to be quasi-isometric to a tree. Using the simplicial boundary, we characterize essential cube complexes for which the contact graph is bounded. We show that the rank-one isometries of $\mathbf X$ that do not virtually stabilize hyperplanes are exactly those isometries of $\mathbf X$ that act with an unbounded quasiconvex orbit on the contact graph. Conversely, we state a form of the Caprace-Sageev rank-rigidity theorem in terms of the contact graph. Using this statement and properties of the simplicial boundary, we study the divergence of groups acting properly and cocompactly on CAT(0) cube complexes, generalizing a result of Behrstock-Charney for right-angled Artin groups.
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