Non-realizability and ending laminations: proof of the density conjecture
type: Item
instance of: scholarly article
title: Non-realizability and ending laminations: proof of the density conjecture
zbMATH Open document ID: 1258.57010
author: Hossein Namazi
author: Juan Souto
published in: Acta Mathematica
publication date: 2013-01-17T00:00:00Z
review text: 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 o…
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 6126651
zbMATH Keywords: Kleinian group
zbMATH Keywords: density conjecture
zbMATH Keywords: ending lamination
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
Wikidata QID: Q61987740
full work available at URL: s11511-012-0088-0
OpenAlex ID: W1978872966
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DOI: 10.1007/S11511-012-0088-0
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