Some virtually special hyperbolic 3-manifold groups

Abstract

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. A variety of attractive properties follow: such manifolds are virtually fibered; their fundamental groups are LERF; and their geometrically finite subgroups 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 of examples with complements not commensurable to any reflection group in H 3 .

Hyperplane sections in arithmetic hyperbolic manifolds

“…In fact, the proof of Theorem 1.4 extends to show that non-cocompact arithmetic lattices virtually retract onto their geometrically finite subgroups. (For instance, see [11,Proposition 1.4]. )…”

Geometrically finite amalgamations of hyperbolic 3‐manifold groups are not LERF

“…In , it is shown that (relatively) quasi‐convex subgroups of virtually compact special (relative) hyperbolic groups are virtual retractions. The celebrated virtually compact special theorem of Agol and Wise ( for cusped manifolds and for closed manifolds) implies that groups of finite volume hyperbolic 3‐manifolds are virtually compact special.…”

“…We will use Agol's construction of virtual fibered structures and the virtual retract property of geometrically finite subgroups to prove the existence of an ‘algebraically fibered’ structure on a subgroup of , with nontrivial induced graph of group structure. The precise statement and its proof is given in Section 3.…”

Separability of double cosets and conjugacy classes in 3-manifold groups

“…Many hyperbolic 3-manifold and orbifold groups are known to be virtually special [7,8,10,11], among them non-cocompact arithmetic and standard cocompact arithmetic lattices. Wise has announced a proof that the fundamental group of any hyperbolic 3-manifold containing an embedded geometrically finite surface is virtually special [31]; the heart of his proof is contained in [32].…”