A boundary criterion for cubulation

  title={A boundary criterion for cubulation},
  author={Nicolas Bergeron and Daniel T. Wise},
  journal={American Journal of Mathematics},
  pages={843 - 859}
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a ${\rm CAT}(0)$ cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered. 
Relative cubulations and groups with a 2-sphere boundary
We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume
Hyperfiniteness of boundary actions of cubulated hyperbolic groups
We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action
Detecting sphere boundaries of hyperbolic groups
We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our
Sublinearly Morse boundaries from the viewpoint of combinatorics
We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes,
Cocompact cubulations of mixed 3-manifolds
In this paper, we complete the classification of which compact 3manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical
Hyperbolic groups with planar boundaries
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually
Hyperbolic groups with planar boundaries
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually
The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are
Hyperbolic groups acting improperly
In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of
Agol's theorem on hyperbolic cubulations
  • Sam Shepherd
  • Mathematics
    Rocky Mountain Journal of Mathematics
  • 2021
The following theorem was proven by Agol: let $G$ be a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex $X$, then $G$ has a finite index subgroup $G'$ that acts freely on $X$


The Structure of Groups with a Quasiconvex Hierarchy
Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows
Codimension-1 Subgroups and Splittings of Groups
Abstract We show that under certain circumstances, a codimension-1 subgroup H of a finitely generated group G either provides a splitting of G as an amalgam or provides a codimension-1 subgroup of H
Criteria for virtual fibering
We prove that an irreducible 3‐manifold with fundamental group that satisfies a certain group‐theoretic property called RFRS is virtually fibered. As a corollary, we show that 3‐dimensional
Special Cube Complexes
Abstract.We introduce and examine a special class of cube complexes. We show that special cube-complexes virtually admit local isometries to the standard 2-complexes of naturally associated
Infinite groups : geometric, combinatorial and dynamical aspects
Parafree Groups.- The Finitary Andrews-Curtis Conjecture.- Cuts in Kahler Groups.- Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries.- Solved and Unsolved Problems Around One
THEOREM 1.1. Let F be an arithmetic lattice in the real Lie group SO(n,1), n > 2. (If n = 7 then assume r does not come from the twisted forms 3'6D4. If n = 3 and F comes from the units of a
Cycles géodésiques transverses dans les variétés hyperboliques
Abstract. Let M be a hyperbolic manifold, with $ \pi_1M $ finitely generated. Let c1 and c2 be two transverse geodesic cycles with dim(c1) + dim(c2) = dim M and $ c_1 \cap c_2 \neq \emptyset $. In
Convergence groups and configuration spaces
We give an account of convergence groups from the point of view of groups which act properly discontinuously on spaces of distinct triples. We give a proof of the equivalence of this characterisation
Immersing almost geodesic surfaces in a closed hyperbolic three manifold
Let M 3 be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into M 3 so that for each, one the corresponding mapping on the universal covering spaces is an
Hyperplane sections in arithmetic hyperbolic manifolds
It is proved that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homological group of the homologists of the manifolds.