# A boundary criterion for cubulation

@article{Bergeron2009ABC, title={A boundary criterion for cubulation}, author={Nicolas Bergeron and Daniel T. Wise}, journal={American Journal of Mathematics}, year={2009}, volume={134}, 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.

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## References

SHOWING 1-10 OF 34 REFERENCES

The Structure of Groups with a Quasiconvex Hierarchy

- Mathematics
- 2009

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

- Mathematics
- 1997

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

- Mathematics
- 2008

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

- MathematicsThe Structure of Groups with a Quasiconvex Hierarchy
- 2008

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

- Mathematics
- 2005

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…

EIGENVALUES OF THE LAPLACIAN, THE FIRST BETTI NUMBER AND THE CONGRUENCE SUBGROUP PROBLEM

- Mathematics
- 1996

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

- Mathematics
- 2002

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

- Mathematics
- 1999

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

- Mathematics
- 2009

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

- MathematicsJ. Lond. Math. Soc.
- 2011

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.