On universal hyperbolic orbifold structures in $S^{3}$ with the Borromean rings as singularity

Abstract

An orientable $3$-orbifold is universal iff every closed, orientable $3$-manifold is the underlying space of an orbifold structure that is an orbifold-covering of it. The first known example of universal orbifold was $\textbf{B}_{4,4,4}=(S^{3}, B,4)$ where $B$ denotes the Borromean rings and all the isotropy groups are cyclic of order 4. The main result in this article is that the hyperbolic orbifold $\textbf{B}_{m,2p,2q}$ is universal for every $m\geq 3$, $p\geq 2$, $q\geq 2$.