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From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-incompressible surfaces) to obtain a collection of $3$--balls. A hyperbolic structure is put on the $3$--balls, and then a bootstrapping argument results in re-gluing the hierarchy components so as to get a hyperbolic structure on the original manifold.

I certainly do not know the details for the above argument, but was wondering if there are any conditions on the hierarchy surfaces needed for the above argument to go through? In other words, are we required to choose a particular hierarchy to ensure that the re-gluing can be done consistently (or can one assume that any hierarchy will do ie- only the existence of a hierarchy is needed)? If a hierarchy with certain properties is needed, what are the conditions on the hierarchy?

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    $\begingroup$ A remark on the proof: the proof in the case of fibered or semi-fibered surfaces (the ``double limit theorem") is much different from the other case (which boils down to the skinning map being contracting plus a bit more). $\endgroup$ – Ian Agol Oct 10 '15 at 20:30
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Yes, there are conditions; basically you want to maintain the hypothesis of not having incompressible annuli at each stage. A really good reference for this is Morgan's essay, "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman, The Smith conjecture (New York, 1979), Pure Appl. Math. 112, Boston, MA: Academic Press, pp. 37–125. I'd suggest looking there.

(I edited out the "tori"; this will be automatic if the original manifold is atoroidal as it should be.)

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  • $\begingroup$ Thanks for the reference. I could be muddling things up, but isn't the conditions of no incompressible tori/annuli a condition on the manifold rather than anything to do with the hierarchy? To put it another way, suppose I have two explicitly given hierarchies $\mathcal{H}_1$ and $\mathcal{H}_2$ for a manifold $M$ that satisfies all conditions for being a hyperbolic $3$--manifold. If I cut up and re-glued each of these in turn, can it happen I end up with a hyperbolic structure on $(M,\mathcal{H}_1)$ but not on $(M,\mathcal{H}_2)$. $\endgroup$ – Don Shanil Oct 11 '15 at 6:29
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    $\begingroup$ If M is closed, the hyperbolic structure is unique, so independent of hierarchy. The argument is inductive, which is where the condition on annuli comes in, to provide the correct topological hypotheses at each stage of cutting along a surface in the hierarchy. As Ian commented above, the argument is different if M fibers over the circle. $\endgroup$ – Danny Ruberman Oct 11 '15 at 12:08

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