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Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $mathbb S^3$. Then, if $M$ admits an exhaustion by $pi_1$-injective sub-manifolds there exists cantor sets $C_nsubset mathbb S^3$ such that $N_n=mathbb S^3setminus C_n$ is hyperbolic and $N_nrightarrow M$ geometrically.
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Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $Phi (c)$, such that if $K$ is a nontrivial knot in the three-sphere with a diagram with $c$ crossings and a particularly simple JSJ decomposition then the complement of $K$ has a finite-sheeted, noncyclic cover with at most $Phi (c)$ sheets.
Given a link in a 3-manifold such that the complement is hyperbolic, we provide two modifications to the link, called the chain move and the switch move, that preserve hyperbolicity of the complement, with only a relatively small number of manifold-link pair exceptions, which are also classified. These modifications provide a substantial increase in the number of known hyperbolic links in the 3-sphere and other 3-manifolds.
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