Virtually Haken surgery conjecture for hyperbolic knot manifolds

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A knot manifold is an irreducible, orientable, compact 3-manifold whose boundary is a single torus. It is hyperbolic if its interior admits a complete hyperbolic metric of finite volume. A slope on a torus is a non-trivial isotopy class of simple closed curves, and M(α)M(\alpha) denotes Dehn filling a 3-manifold MM along a slope α\alpha on its torus boundary. A knot manifold has Property VH if M(α)M(\alpha) is virtually Haken for all but finitely many slopes α\alpha on ∂M\partial M. Virtually Haken surgery conjecture. Let MM be a hyperbolic knot manifold. Then MM has Property VH. This is identified as an important special case of Waldhausen's conjecture; the source does not state whether it has been resolved.

References

Primary source

Joseph D. Masters, “Virtually Haken surgeries on once-punctured torus bundles”, arXiv:math/0506443 (2006).

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