Non-realizability and ending laminations: proof of the density conjecture
One of the classical conjectures for Kleinian groups is the Bers-Sullivan-Thurston density conjecture which states that every finitely generated Kleinian group is an algebraic limit of geometrically finite groups. In its original form it was formulated by Bers who asked if every Kleinian group isomorphic to a surface group is an algebraic limit of quasi-Fuchsian groups; for groups without parabolic elements, this original conjecture was proved by \textit{K. W. Bromberg} [Ann. Math. (2) 166, No. 1, 77--93 (2007; Zbl 1137.30014)], using the deformation theory of cone-manifolds, and then for freely indecomposable Kleinian groups by \textit{J. F. Brock} and \textit{K. W. Bromberg} [Acta Math. 192, No. 1, 33--93 (2004; Zbl 1055.57020)]. In the present paper, the density conjecture is proved for the general case. The solution depends on the recent proofs of the tameness conjecture by Agol and Calegari-Gabai, and the ending lamination conjecture by Brock-Canary-Minsky. The ending lamination conjecture states that a hyperbolic 3-manifold \(N\) uniformized by a f.g. Kleinian group is determined up to isometry by its topological type and its end invariants (a conformal structure on the surface at infinity associated to each geometrically tame end, and an ending lamination on the surface \(F\) associated to a degenerate, i.e. not geometrically tame end); the tameness conjecture instead states that each end is topologically tame, that is homeomorphic to the product of a surface \(F\) with a ray. ``We say that an ending lamination \(\lambda\) is \textit{realized} in such a hyperbolic 3-manifold \(N\) if there exists a finite-area complete hyperbolic metric \(\sigma\) on \(F\) and a proper map \(f:(F,\sigma) \to N\) which is totally geodesic on leaves of \(\lambda\) and which, on the level of fundamental groups, induces the same map as the embedding \(F \hookrightarrow N\). An important property of the ending lamination of an end of \(N\) is that it is not realized, by a result of Canary. Our proof of the density conjecture basically amounts to proving that in some sense this property identifies the ending lamination. This is in fact the most important contribution of this work to the density conjecture.'' Another approach to a complete proof of the density conjecture using the deformation theory of hyperbolic cone-manifolds, which does not contain any reference to the ending lamination conjecture, has been announced by Bromberg and Souto (but the solution of the tameness conjecture remains essential).
scientific article; zbMATH DE number 6126651
scientific article