A Cantor set with hyperbolic complement (Q2841088)

The authors construct a (non-trivial) embedding of the Cantor set into the three-dimensional sphere \(\mathbb{S}^3\) such that the complement of the image admits a complete hyperbolic structure. This Cantor set is approximated by the union of finitely many disjoint \(\theta\)-graphs in \(\mathbb{S}^3\) (as the Antoine's necklace is approximated by the union of finitely many disjoint circles).NEWLINENEWLINETo show the existence of a hyperbolic metric, the authors give a sufficient condition for an open 3-manifold \(M\) to admit a complete hyperbolic metric. Namely, a sufficient condition is the existence of a nested exhaustion \(M = \bigcup_{n = 1}^\infty K_n\), where \(K_n\) is a sub-manifold of \(M\) with boundary such that \(K_n \setminus K_{n-1}\) is acylindrical and the genera of the boundary components of the submanifolds \(K_n\) are bounded.NEWLINENEWLINEThey also ask whether the hyperbolic metric of the complement of such a Cantor set is unique.