Branched covers over strongly amphicheiral links

Abstract

THIS PAPER deals with the question: which 3-manifolds have fundamental groups which are virtually Z-representable-have a subgroup of finite index which maps onto the infinite cyclic group, Z. A compact, orientable, irreducible 3-manifold with this property is finitely covered by a Haken manifold (is virtually Haken). The Haken manifolds are quite well understood: they are determined by homotopy type (of the pair (M, 8M)) [19], they have a solvable classification problem [4], [6], they have residually finite fundamental groups [lo], and they have a unique torus-annulus decomposition [15], [16] into geometric pieces [lS]. There is hope in extending much of these results for Haken manifolds to virtually Haken manifolds. It has been conjectured that for a 3-manifold, M, z,(M) is infinite if and only if xl(M) is virtually Z-representable (see [7], [8] for more discussion). In this paper we will consider (closed, orientable) 3-manifolds which are described as branched covers of the 3-sphere, branched over a fixed link, L c S3. There are many known examples of universal links-for which every 3-manifold can be represented as a cover of S3 branched over the link. Each 2-bridge knot or link, with the exception of those which are also torus knots or links, is universal [14]. For any link, L, in a 3-manifold, M, the collection of coverings of M branched over L forms a lattice. We observe (Lemma 2.3) that whenever an element of this lattice has virtually Z-representable fundamental group then so does every element above it in the lattice. We combine this with an analysis of the upper bound of a pair of elements (Theorem 2.1) to study how the property of virtual Z-representability proliferates throughout the lattice. When the link is strongly amphicheiral (invariant under an orientation reversing involution of S3), the involution theorem (section 3) will apply to show that certain coverings branched over the link will have virtually Z-representable fundamental group. Others will inherit this property by comparison. This includes those whose branching indices satisfy certain divisibility conditions (Theorem 4.1). We apply this to some of the known strongly amphicheiral universal links such as the figure eight knot and the Borromean rings. For example, any cover of S3 branched over the figure eight knot all of whose branching indices are divisible by some integer q > 2 has virtually Z-representable fundamental group (Corollary 4.3). This improves on results of [1], [12]. Throughout this paper we will consider only closed, orientable 3-manifolds. A map