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What is the status of the following conjecture:

"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one obtains a manifold whose shortest closed geodesic has length $2.12255\ldots$ with angle of twist $\pm 1.80911 \ldots,$ and has self-replicating behaviour when removed."

This is Conjecture 6.1 from p. 2568 of Thurston's expository paper "How to see 3-manifolds" (Class. Quantum Grav. 15 (1998) 2545--2571).

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I think this is still open. – Ian Agol Mar 15 '13 at 4:06
Adams' paper on simple geodesics has a heuristic explanation of why this might be happening. – Robert Haraway Aug 3 '13 at 1:39

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