Some virtually special hyperbolic 3-manifold groups

TL;DR

This paper proves that certain hyperbolic 3-manifolds with right-angled ideal polyhedral decompositions are virtually fibered and have fundamental groups with desirable properties, extending previous results to a broader class.

Contribution

It generalizes Haglund and Wise's theorem to relatively hyperbolic groups and shows these manifolds are virtually fibered with LERF fundamental groups.

Findings

Manifolds admit a virtually special square complex deformation retraction.

Fundamental groups are LERF and geometrically finite subgroups are virtual retracts.

Classified low-complexity augmented links and identified an infinite family with unique properties.

Abstract

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.