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Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives. (English) Zbl 1061.30037
With his hyperbolic Dehn surgery theorem and later the orbifold theorem, Thurston demonstrated the power of using hyperbolic cone-manifolds to understand complete, non-singular hyperbolic 3-manifolds. In this paper, the author uses Hodgson and Kerckhoff’s techniques to study infinite volume hyperbolic 3-manifolds. The obtained results have many applications: the Bers density conjecture, the density of cusps on the boundary of quasiconformal deformations spaces, and for constructing type preserving sequences of Kleinian groups. The simplest example of the studied problem is the following: Let \(M\) be a hyperbolic 3-manifold and \(c\) a simple closed geodesic in \(M\). Then the topological manifold \(M\setminus c\) also has a complete hyperbolic metric \(\widehat M\). How does the geometry of \(M\) relate to that of \(\widehat M\)? Another question is the following: Assume \(\Gamma\) is a Kleinian group and \(\Gamma_i\) is a sequence of geometrically finite Kleinian groups such that \(\Gamma_i\to\Gamma\), algebraically. Does there exist a type preserving sequence \(\Gamma_i'\) of geometrically finite groups also converging to \(\Gamma\)? Here type preserving means that if elements \(\gamma_i\) converge to \(\gamma\), then \(\gamma\) is parabolic if and only if the \(\gamma_i\) are parabolic. The starting point is the local parametrization of hyperbolic cone-manifolds. The author describes some of Hodgson and Kerckhoff’s results on tubes in cone-manifolds and estimates on the radius of embedded tubes about the cone-singularity, and uses these ideas to show that embedded hyperbolic half spaces are disjoint from these tubular neighborhoods of the cone-singularity. The analytic deformation theory of cone-manifolds is used to control the length of geodesies as the cone-angle decreases. The key result of the paper is the following: The \(L^2\)-norm of the cone-manifold deformation bounds the change in projective structure. First this is done for a hyperbolic half space and then a large embedded half space is found in each geometrically finite end which allows one to globally bound the deformation of the entire projective structure. The final step is to understand the geometric limit of a sequence of cone-manifolds. Given a geometrically finite cone-manifold, under certain conditions, there is a one-parameter family of cone-manifolds decreasing the cone-angle to zero.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30-06 Proceedings, conferences, collections, etc. pertaining to functions of a complex variable
57M50 General geometric structures on low-dimensional manifolds
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