Some virtually special hyperbolic 3-manifold groups

Details Some virtually special hyperbolic 3-manifold groups Some virtually special hyperbolic 3-manifold groups

2009-03-30

arxiv.org/abs/0903.5288v3

Mathematics

Geometric Topology

51 pages, 13 figures. Referee's comments incorporated. To appear in Commentarii Mathematici Helvetici

Scientific paper

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem of Haglund and Wise to the relatively hyperbolic setting and deduce that the fundamental group of M is LERF and that the geometrically finite subgroups of the fundamental group are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family with complements not commensurable to any 3-dimensional reflection orbifold.

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