Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries

摘要

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 partial derivative(Delta) 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 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 X to boundedness of the 1-skeleton of partial derivative(Delta) 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, partial derivative(Delta) X. Finally, we construct, for each n >= 0, k >= 2, 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.