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A boundary criterion for cubulation. (English) Zbl 1279.20051

Summary: We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a 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 J. Kahn and V. Marković [Ann. Math. (2) 175, No. 3, 1127-1190 (2012; Zbl 1254.57014)], and I. Agol’s criterion [J. Topol. 1, No. 2, 269-284 (2008; Zbl 1148.57023)], we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
57M50 General geometric structures on low-dimensional manifolds
57M10 Covering spaces and low-dimensional topology
55R05 Fiber spaces in algebraic topology
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