Hyperbolic Structures on Branched Covers over Hyperbolic Links

Using a result of Tian concerning deformation of negatively curved metrics to Einstein metrics, we conclude that, for any fixed link with hyperbolic complement, there is a class of irregular branched covering spaces, branched over that link, effectively detectable by their branching indices, which consists entirely of closed hyperbolic manifolds. Section 0 Introduction. One of the elusive components of the Thurston Geometrization Conjecture is the the conjecture that all 3-dimensional closed manifolds with a Riemannian metric of negative sectional curvature admit a Riemannian metric of constant negative sectional curvature – the “negatively curved implies hyperbolic” conjecture. This conjecture is known to be false in dimensions higher than 3 (see [G-Th]), but is still very much a viable conjecture in dimension 3 (and is of course true in dimension 2). One of the easiest ways to construct 3-manifolds admitting negatively curved metrics is to take a hyperbolic orbifold with singular locus a link and consider sufficiently branched covers over this orbifold, that is, branched covers whose branching indices over a given component of the singular locus of the orbifold are greater than or equal to the order of the isotropy group of that component. These are precisely the branched covers on which the cone manifold structure lifted from the base orbifold has all cone angles > 2π (see [Ho] and [Jo] for discussions of cone manifolds). These cone manifold structures are singular metrics which may be smoothed to Riemannian metrics of negative sectional curvature. This construction is of particular importance to 3-dimensional topology in light of the existence of universal links, that is, links such that all closed, orientable 3-manifolds are obtainable as branched covers over that fixed link. Furthermore, all universal links are the singular locus of a hyperbolic orbifold, so this construction potentially has considerable bearing on the classification problem for 3-manifolds. The purpose of this paper is to combine this smoothing technique, by way of some explicit pinching constant computations, with a result of Tian (see [Ti]) to show Theorem 3.2. Let (M,L) be a 3-manifold and a link such that M−L admits a hyperbolic metric of finite volume. Let L1, . . . , Lq be the components of L. Let (M̂, L̂) be a branched cover over (M,L) with minimum branching index ni over Li and maximum branching index Ni over Li. Denote by Qi the number of components of branching locus over Li with 1991 Mathematics Subject Classification. Primary: 57R15, Secondary: 57M12, 53C25.