In [4], Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. He, moreover, proved that the size of the resolution is bounded by a constant depending only on the group. Extending Dunwoody's definition of patterns we construct resolutions for group actions on a general finite dimensional CAT(0) cube complex. In dimension two, we bound the number of hyperplanes of this resolution. We apply this result for surfaces and 3-manifolds to bound collections of codimension-1 submanifolds.Comment: 17 pages, 10 figure

We describe a higher dimensional analogue of Stallings' folding sequences for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups that act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.

We show that a one-ended simply connected at infinity hyperbolic group G with enough codimension-1 surface subgroups has ∂G ∼ = S 2 . By Markovic [16], our result gives a new characterization of virtually fundamental groups of hyperbolic 3-manifolds.We begin by a survey of definition and results concerning CAT(0) cube complexes and quasiconvex subgroups of hyperbolic groups. For a more complete survey see, for example, Sageev [18]. 2.1. CAT(0) cube complexes and hyperplanes. Definition 2.1. [CAT(0) cube complexes] A cube complex is a complex made by gluing unit Euclidean cubes (of varying dimension) along their faces using isometries. A cube complex is CAT(0) if it is CAT(0) with respect to the quotient metric induced by endowing each cube with the Euclidean metric (See [7]).As was first observed by Sageev [17], CAT(0) cube complexes naturally carry a combinatorial structure given by the associated hyperplanes and halfspaces. We now recall their definition and properties.Definition 2.2.[hyperplanes] Let X be a CAT(0) cube complex. The equivalence relation on the edges of X generated by e ∼ e when e and e are parallel edges in a square of X is called the parallelism relation. The equivalence classes of edges under the parallelism relation are the combinatorial hyperplanes of X. The convex hull of the midpoints of the edges of a combinatorial hyperplane is called a hyperplane. We denote the set of hyperplanes in X byĤ =Ĥ(X).The main features of hyperplanes are summed in the following. Proposition 2.3. Let X be a CAT(0) cube complex. Then every hyperplaneĥ ∈Ĥ is naturally a CAT(0) cube complex of codimension-1 in X, and X \ĥ has exactly two components.Definition 2.4. The components of X \ĥ are the halfspaces of X associated to (or bounded by)ĥ. The set of halfspaces is denoted by H = H(X). There is a natural mapˆ: H →Ĥ which maps each halfspace to its bounding hyperplane. The set of halfspaces also carries a natural complementation involution * : H → H which maps each halfspace h to the other component of X \ĥ. 2.2.Cubulating hyperbolic groups. Recall the following definitions about quasiconvex subgroups of Gromov hyperbolic groups.Definition 2.5. A quasiconvex subgroup H of a hyperbolic group G is codimension-1 if G/H has more than one end. The group G has enough codimension-1 subgroups if every two distinct points in ∂G can be separated by the limit set of a codimension-1 quasiconvex subgroup.By results of Sageev [17], Gitik-Mitra-Rips-Sageev [10] and Bergeron-Wise [3] we have the following.Theorem 2.6. Let G be a hyperbolic group with enough codimension-1 subgroups, then G acts properly cocompactly on a finite dimensional CAT(0) cube complex whose hyperplane stabilizers belong to the family of codimension-1 subgroups.

We bound the size of $d$-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.Comment: 12 page

We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.