Embeddability of noncompact hyperbolic manifolds as complements of codimension-1 and -2 submanifolds

Abstract

We consider the question of when a finite volume hyperbolic ( n+1)-manifold M may be embedded as a complement of a closed codimension- k submanifold A inside a closed ( n+1)-manifold N. This is possible if and only if every flat manifold E corresponding to an end of M is either an S 0- or S 1-bundle, i.e., k is either 1 or 2. We give criteria when a flat manifold E is an S 0- or S 1-bundle and use them to determine which of the 10 flat 3-manifolds are S 0- or S 1-bundles. Also, we construct examples of flat manifolds in every dimension ⩾3 that are not S 0- or S 1-bundles, showing there are obstacles, in general, to viewing hyperbolic manifolds as complements. Furthermore, in contrast to the 3-dimensional case, where there are infinitely many knot complements with a hyperbolic structure, we show that there are at most finitely many 4-manifolds M that are codimension-2 complements inside any fixed N (e.g., the 4-sphere). If M is a codimension-1 complement, we show that the universal cover of N is R n+1 and, with an additional assumption, that for a given M there are only finitely many choices for N. It should be noted that some of the results involving dimension 4 depend on an extension of a theorem of Farrell and Hsiang which was asserted to be true without proof by Quinn [Math. Rev. 84k (1984) 57017]. Also, some of the results involving dimension 3 require the assumption of irreducibility.