In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--manifolds. Now, one consequence of this is the theorem stated in the paper, where he provides a brief outline showing that if $M$ is a Haken $3$--manifold, with hyperbolic structure of finite volume ($M$ is compact irreducible with torus boundary components) then there exists a decomposition of $M$ into manifolds $(M_{1},\ldots,M_{k})$ with $M_{k} = T^{2}\times I$ such that for each $i$, $M_{i}$ is hyperbolic.

Now suppose I have a hierarchy $\mathcal{H}$ for a Haken $3$--manifold $M$, where $M$ is compact,irreducible and has torus boundary. Let $\mathcal{H} = (M_{1},\ldots,M_{k})$ and suppose manifold $M_{k} = T^2\times I$. Would any one know the conditions required of $\mathcal{H}$ to satisfy Gabai's theorem mentioned above. I guess another method of looking at this is to ask for the obstruction to proving the converse of Gaba's theorem?

PS. As Neil has suggested I also assume that $M$ is atoroidal (required in the context of Thurston and hyperbolization).

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    $\begingroup$ Are you assuming that the interior of $M$ is atoroidal? $\endgroup$ – Neil Hoffman Aug 13 '13 at 8:45
  • $\begingroup$ Yes, I also assume that $M$ is atoroidal, thanks Neil for pointing that out. $\endgroup$ – Don Shanil Aug 14 '13 at 4:06
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    $\begingroup$ This is by no means an answer, but you might consider Lackenby's paper "Taut ideal triangulations" (G&T 2000). I believe there's a relationship between those triangulations and angle structures (if not actual hyperbolic structures) and by construction the taut ideal triangulation is constructed from taut id. triang. on the pieces at the end of the hierarchy. $\endgroup$ – Scott Taylor Aug 14 '13 at 16:38
  • $\begingroup$ Yes, Having an angle structure is good enough for me. The motivation for the question is that I can construct a class of hierarchies for the manifold in question and was hoping to construct an angle structure using this (or at least to get some condition when this is possible). $\endgroup$ – Don Shanil Aug 15 '13 at 5:59
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    $\begingroup$ In Theorem 1, the "Internal Hierarchy" may terminate in one of 6 possibilities, one of which is $T^2\times I$. But there is no hyperbolicity statement there. In Theorem 2, one interpolates between finite-volume hyperbolic manifolds, keeping everything hyperbolic in between. However, $T^2\times I$ does not have a finite-volume hyperbolic metric. So I'm confused which theorem you're referring to? $\endgroup$ – Ian Agol Aug 22 '13 at 3:16

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