Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

Jason Behrstock, Mark F. Hagen

Research output: Contribution to journalArticle (Academic Journal)peer-review

10 Citations (Scopus)
179 Downloads (Pure)


Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of P. As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cube complex X to boundedness of the 1-skeleton of the boundary of X. We deduce characterizations of thickness and strong algebraic thickness of a group G acting properly and cocompactly on the CAT(0) cube complex X in terms of the structure of, and nature of the G-action on, the boundary of X. Finally, we construct, for each n,k, infinitely many quasi-isometry types of group G such that G is strongly algebraically thick of order n, has polynomial divergence of order n+1, and acts properly and cocompactly on a k-dimensional CAT(0) cube complex.
Original languageUndefined/Unknown
Pages (from-to)649-707
JournalGroups, Geometry and Dynamics
Issue number2
Early online date9 Jun 2016
Publication statusPublished - 2016

Bibliographical note

Corrections according to referee report. Fixed proof of Theorem 4.3. To appear in "Groups, Geometry, and Dynamics"


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