On universal hyperbolic orbifold structures in S3 with the Borromean rings as singularity

Abstract

A link in S3 is called a universal link if every closed orientable 3-manifold is a branched cover of S3 over this link. It is well known that the Borromean rings and many other links are universal links. The question whether a link is universal can be naturally extended to orbifolds. An orbifold M is said to be universal if every closed orientable 3-manifold is the underlying space of an orbifold which is an orbifold covering of M. Let Bm,n,p denote the orbifold whose underlying space is S3, whose singular set is the Borromean rings B, and whose isotropy groups for the three components of B are cyclic groups of orders m, n and p. In an earlier paper of H. M. Hilden et al. [Invent. Math. 87 (1987), no. 3, 441–456;], it was shown that B4,4,4 is universal. In this paper, the authors generalize this result and prove that Bm,2p,2q is universal for every m≥3, p≥2, q≥2.