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On the density of geometrically finite Kleinian groups. (English) Zbl 1055.57020
Acta Math. 192, No. 1, 33-93 (2004); erratum ibid. 219, No. 1, 17-19 (2017).
According to the Bers-Sullivan-Thurston density conjecture, a complete hyperbolic 3-manifold \(M\) with finitely generated fundamental group is a limit of geometrically finite hyperbolic 3-manifolds.
The main result of this highly technical and elegant paper is the following. Let \(M\) be a complete hyperbolic 3-manifold with finitely generated fundamental group, incompressible ends and no cusps. Then \(M\) is an algebraic limit of geometrically finite hyperbolic 3-manifolds.
A manifold \(M\) is said to be geometrically finite if its convex core, the minimal convex subset of \(M\), has finite volume. \(M = \mathbb H^3 / \Gamma\) is an algebraic limit of \(M_i = \mathbb H^3 / \Gamma_i\) if there are isomorphisms \(\rho_i : \Gamma \to \Gamma_i\) so that after conjugating the groups \(\Gamma_i\) in \(\text{ Isom}^{+} (\mathbb H^3)\) if necessary, we have \(\rho_i(\gamma) \to \gamma\) for each \(\gamma \in \Gamma\). \(M\) has incompressible ends if it is homotopy equivalent to a compact submanifold with incompressible boundary. To obtain the result, the authors develop the deformation theory of 3-dimensional hyperbolic cone-manifolds and prove the drilling theorem. The principal application of the cone-deformation theory is its ability to control the geometric effect of a cone-deformation that decreases the cone-angle at the singular locus when the singular locus is a sufficiently short geodesic.

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