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Geometrically finite Kleinian groups and parabolic elements. (English) Zbl 0903.30032
The purpose of this paper is to prove the following statement. Let \(\Gamma\) be a torsion-free Kleinian group. We suppose that we are given a set of non-conjugate loxodromic primitive elements of \(\Gamma\) (the non-conjugacy also applies to the inverses) corresponding to disjoint curves on the boundary then we can deform the \(\Gamma\) to a geometrically finite Kleinian group on the boundary of space of quasiconformal deformations in such a way that the set converges to a set of parabolic elements. This remarkable theorem extends earlier results of Maskit and the author. The proof makes use of Thurston’s approach to the deformation theory of Kleinian groups and in particular of an apparently unpublished compactness (convergence) theorem due to Thurston. The author announces that he has found an alternative approach using Thurston’s uniformization theorem.

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