Geometric models for hyperbolic 3-manifolds

In [Mi4,BrocCM1], Minsky, Brock and Canary gave a proof of Thurston’s Ending Lamination Conjecture for indecomposable hyperbolic 3-manifolds. In this paper, we offer another approach to this which was inspired by the original. Many of the key results are similar, though the overall logic is somewhat different. A possible advantage is that much of the relevant theory of “hierarchies” and other, more technical, parts of the proof can be simplified. We hope that it will help to render some of these ideas more generally accessible. In another paper [Bow5], we show how to adapt these arguments to deal with the general (decomposible) case of a tame 3-manifold. Brock, Canary and Minsky have also stated that they can similarly adapt argument in this regard. Some brief remarks to this effect are given in [BrocCM1], and a sequel is promised as [BrocCM2]. We also note that an alternative approach to the Ending Lamination Conjecture has recently been proposed by Brock, Bromberg, Evans and Souto [BrocBES]. Their approach does not involve the construction of a model. In the late 1970s Thurston made several important conjectures relating to the geometry of 3-manifolds. Probably the most famous is the “Geometrisation Conjecture” recently announced by Perelman, but the “Tameness Conjecture” and “Ending Lamination Conjecture”, have also been central to the development of hyperbolic geometry, and together amount to a complete classification of finitely generated torsion free Kleinian groups. Let M be a complete hyperbolic 3-manifold with π1(M) finitely generated. The Tameness Conjecture classifies the “geometric” ends of M into two sorts “geometrically finite” and “simply degenerate”. This was shown by [T,Bon,Can] to be equivalent to Marden’s earlier conjecture [Mar], namely that M is topologically finite, i.e. the interior of a compact manifold. Tameness was proven in the “indecomposable” case by Bonahon [Bon], and recently in general by Agol [Ag] and by Calegari and Gabai [CalG] (see also [So] and [Bow6]). The Ending Lamination Conjecture goes on to say that M is completely determined up to isometry by a finite set of “end invariants”. A number of special cases of this conjecture were established by Minsky [Mi1,Mi2,Mi3], and a proof for all indecomposable 3-manifolds is given in [Mi4,BroCM1]. This makes much use of the work of Masur and Minsky on the curve complex, in particular, hyperbolicity [MasM1] and the theory of tight geodesics and hierarchies [MasM2]. We should note that this work has had implications beyond the theory of 3-manifolds. The Ending Lamination Conjecture is closely related to the large scale structure of Teichmüller space and other geometries associated to a surface, and hence has applications