Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

  • Jason Behrstock

    Lehman College, CUNY, Bronx, USA
  • Mark F. Hagen

    University of Cambridge, UK


Let be a group acting geometrically on a CAT(0) cube complex . We prove first that is hyperbolic relative to the collection of subgroups if and only if the simplicial boundary 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 . As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska and 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 to boundedness of the 1-skeleton of . We deduce characterizations of thickness and strong algebraic thickness of a group acting properly and cocompactly on the CAT(0) cube complex in terms of the structure of, and nature of the -action on, . Finally, we construct, for each , infinitely many quasi-isometry types of group such that is strongly algebraically thick of order , has polynomial divergence of order , and acts properly and cocompactly on a -dimensional CAT(0) cube complex.

Cite this article

Jason Behrstock, Mark F. Hagen, Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries. Groups Geom. Dyn. 10 (2016), no. 2, pp. 649–707

DOI 10.4171/GGD/360