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The classification of Kleinian surface groups. II: The Ending lamination conjecture. (English) Zbl 1253.57009
Thurston’s Ending Lamination Conjecture states that a hyperbolic 3-manifold $$M = \mathbb H^3/G$$ with finitely generated fundamental group is determined, up to isometry, by its topological type and its end invariants. This is related also to Marden’s Tameness Conjecture which has been recently proved by Agol and Calegari-Gabai and states that each end of $$M$$ is topologically tame, i.e. homeomorphic to the product $$S \times [0, \infty)$$ of a surface $$S$$ with the positive reals (and hence $$M$$ is homeomorphic to the interior of a compact 3-manifold). The end invariant associated to such a tame end is then a conformal structure on the surface $$S$$ (a point in the Teichmüller space of $$S$$) if the end is also geometrically tame (geometrically finite), and a geodesic lamination on $$S$$ if it is geometrically infinite (but still topologically tame).
In the present paper, the Ending Lamination Conjecture is proved for the basic case of a Kleinian surface group $$G$$ (i.e., $$G$$ is isomorphic to the fundamental group of a compact surface). By the proof of the tameness conjecture in this case (due to Thurston and Bonahon), the associated hyperbolic 3-manifold $$M = \mathbb H^3/G$$ is homeomorphic to a product $$\text{int}(S) \times \mathbb R$$, for a compact orientable surface $$S$$, so there are exactly two end invariants in this case: each is a geodesic lamination on some (possibly empty) subsurface of $$S$$ and a conformal structure on the complementary surface. This implies then also a proof of the Ending Lamination Conjecture for incompressible ends of a general hyperbolic 3-manifold $$M$$ as above. The first part of the proof of the Ending Lamination Conjecture for Kleinian surface groups appeared in a previous paper of Y. Minsky [Ann. Math. (2) 171, No. 1, 1–107 (2010; Zbl 1193.30063)], and the proof of the Ending Lamination Conjecture for the general case of not necessarily incompressible ends will appear in the third paper of the series (in the meantime, Bowditch, Rees and Soma have given alternate proofs of the Ending Lamination Conjecture in which various aspects have been simplified).
As explained in the present paper, the solution of the Ending Lamination Conjecture is also a crucial ingredient in a proof of the last of the three great conjectures on Kleinian groups, the Bers-Sullivan-Thurston Density Conjecture, in particular the results of the present paper imply this conjecture for Kleinian surface groups and for the case of incompressible ends.
Concerning the proofs, “The main technical result that leads to the Ending Lamination Theorem is the Bilipschitz Model Theorem, which gives a bilipschitz homeomorphism from a model manifold $$M_\nu$$ to the hyperbolic manifold $$M = \mathbb H^3/\rho(\pi_1(S))$$. The model $$M_\nu$$ was constructed in the paper by Minsky cited above, and its crucial property is that it depends only on the two end invariants and not on the representation $$\rho: \pi_1(S) \to \text{PSL}_2(\mathbb C)$$ itself.”

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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