A boundary criterion for cubulation

By Nicolas Bergeron, Daniel T. Wise. 2009

164 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/431048bf3bc3c1c44c9944568a2b64f349fcd8e5

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Citations in the corpus

On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, methodology): “ in [24] that if a group is “virtually special” then it satisﬁes both properties. So we need to check that any cocompact group Γ containing π1(N′ α) is so. To do this last thing we use the results of [26] and [8]: accordingto their Theorem 1.4 and Theorem 1.1 respectively it suﬃces to verify that th”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, methodology): “er action on a cube complex from a suﬃciently rich collection of hierarchically quasiconvex codimension–1 subgroups by a method exactly analogous to that used to cubulate various hyperbolic groups in [BW13]. The main diﬀerence is that Gdoes not act as a uniform convergence group on BpG,Sq; one must ”
The Thurston norm and twisted Alexander polynomials (2012, methodology): “njecture 1.3. Let Nbe an irreducible 3-manifold which is not a graph manifold, then π1(N) is virtually RFRS. Partial results have been obtained by Agol [Ag08, Corollary 2.3] [Ag09], Bergeron and Wise [BW09] and Chesebro, Deblois and Wilton [CDW09]. Most importantly, the following result was announce”
Finiteness properties of cubulated groups (2012, methodology): “The idea in [BW12] is to use ∂G to choose finitely many quasiconvex codimension–1 subgroups so that each geodesic (and hence each axis) is separated by a wall.”
The asymptotic growth of torsion homology for arithmetic groups (2010, background): “Since it follows from [5, 6] that arithmetic hyperbolic groups with a nonzero first Betti number are virtual subgroups of (abstract) right angled reflection groups, one gets:19 Arithmetic lattices of SL2(C) are good.”
3-Manifold Groups (2012, unknown): “the years 2007–2012 has lead to the following theorem, which was proved by Agol [Ag12], Kahn–Markovic [KM12] and Wise [Wis12a], with major contributions from Agol–Groves–Manning [Ag12], Bergeron–Wise [BeW12], Haglund–Wise [HaW08, HaW12], Hsu–Wise [HsW12] and Sageev [Sag95, Sag97]. Theorem 5.4. (Virt”