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?



Conditions on the hierarchy for Thurston's hyperbolization theorem
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.)


