Branched covers over strongly amphicheiral links (Q917083)

A group G is called virtually \({\mathbb{Z}}\)-representable if G has a subgroup H with finite index which maps onto the infinite cyclic group \({\mathbb{Z}}\). For a compact orientable irreducible 3-manifold M, if \(\pi_ 1(M)\) is virtually \({\mathbb{Z}}\)-representable, then M is virtually Haken, i.e. M is finitely covered by a Haken manifold. It is expected (but not proven yet) that many nice results for Haken manifolds hold alo for virtually Haken manifolds. The author shows as a consequence of the main theorems, the following theorem. If \(L_ 1\) is the Borromean ring and the branched cover is divisible by some \((q_ 1,q_ 2,q_ 3)\), with at least two components greater than one, or if \(L_ 2\) is the figure eight knot and the branched cover is divisible by some \(q>2\), then the branched cover of \(L_ i\) has virtually \({\mathbb{Z}}\)-representable fundamental group. Here, the branch cover p: \(M\to S^ 3\) along a link L of n components \(K_ i\) is said to be divisible by \(q=(q_ 1,...,q_ n)\) if each branching index of p at each component of \(p^{-1}(K_ i)\) is divisible by \(q_ i\). Note that these links \(L_ i\) \((i=1,2)\) are strongly amphicheiral and universal, i.e. every closed orientable 3-manifold is a finite branched cover of \(L_ i\).