The classification of Kleinian surface groups. I: Models and bounds. (English) Zbl 1193.30063
This is the first of three papers dedicated to a proof of Thurston’s ending lamination conjecture: a hyperbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants (Riemann surfaces or points in Teichmüller spaces for geometrically finite ends, ending laminations for geometrically infinite ones). If all ends are geometrically finite, the conjecture follows from the classical quasiconformal deformation theory of Kleinian groups of Ahlfors and Bers. When an end is geometrically infinite instead, there are two cases. If the boundary of the compact core of the manifold is incompressible, then work of Thurston and Bonahon gives a geometric and topological description of an end (which, in particular, is topologically tame, i.e., a product of a surface and an interval), and allows the ending lamination to be defined. If the core is compressible, the corresponding question of tameness of an end was resolved only recently by Agol and Calegari-Gabai (“the tameness conjecture”).
In the present paper, the incompressible boundary case is considered; by restriction to boundary subgroups, this reduces to the case of Kleinian surface groups. Very roughly, the idea is then to construct a model manifold depending only on the end invariants, together with a bilipschitz homeomorphism from the model to the manifold given by the Kleinian surface group. For two Kleinian surface groups with the same end invariants, this would result then in a bilipschitz homeomorphism between the two corresponding hyperbolic 3-manifolds which, by Sullivan’s rigidity theorem, could be deformed to an isometry. In the present paper, the model manifold together with a map satisfying some Lipschitz bounds is constructed. In the second paper of the series (together with Brock and Canary), this map will then be promoted to a bilipschitz homeomorphism, and finally, in the third paper, the case of a compact core with compressible boundary is considered.
A first version of the present paper has been submitted in 2003 and, as the author notes, in the meantime a number of other proofs of the ending lamination conjecture, including the compressible boundary case, have appeared, independently by Bowditch, Rees, and Soma. This finally resolves the last of the three major central conjectures in the theory of Kleinian groups, the tameness conjecture, the density conjecture, and the ending lamination conjecture, which had their origin in the classical theory of Kleinian groups as promoted by Ahlfors and Bers, and have been made accessible and finally resolved starting with the work of Thurston.
In the present paper, the incompressible boundary case is considered; by restriction to boundary subgroups, this reduces to the case of Kleinian surface groups. Very roughly, the idea is then to construct a model manifold depending only on the end invariants, together with a bilipschitz homeomorphism from the model to the manifold given by the Kleinian surface group. For two Kleinian surface groups with the same end invariants, this would result then in a bilipschitz homeomorphism between the two corresponding hyperbolic 3-manifolds which, by Sullivan’s rigidity theorem, could be deformed to an isometry. In the present paper, the model manifold together with a map satisfying some Lipschitz bounds is constructed. In the second paper of the series (together with Brock and Canary), this map will then be promoted to a bilipschitz homeomorphism, and finally, in the third paper, the case of a compact core with compressible boundary is considered.
A first version of the present paper has been submitted in 2003 and, as the author notes, in the meantime a number of other proofs of the ending lamination conjecture, including the compressible boundary case, have appeared, independently by Bowditch, Rees, and Soma. This finally resolves the last of the three major central conjectures in the theory of Kleinian groups, the tameness conjecture, the density conjecture, and the ending lamination conjecture, which had their origin in the classical theory of Kleinian groups as promoted by Ahlfors and Bers, and have been made accessible and finally resolved starting with the work of Thurston.
Reviewer: Bruno Zimmermann (Trieste)
MSC:
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 57M50 | General geometric structures on low-dimensional manifolds |
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