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Ending laminations and boundaries for deformation spaces of Kleinian groups. (English) Zbl 0715.30032
Let $$\Gamma$$ be a finitely generated, torsion-free Kleinian group satisfying the condition (*) introduced by Bonahon; i.e. if $$\Gamma =A*B$$ for nontrivial subgroups A and B, then there exists a parabolic element of $$\Gamma$$ none of whose conjugates are contained in A or B. Let $$\Omega_{\Gamma}$$ denote the domain of discontinuity of $$\Gamma$$. Consider a measured lamination $$\lambda$$ on $$\Omega_{\Gamma}/\Gamma$$. The author gives a sufficient (actually also necessary) condition so that there exists an algebraic limit (G,$$\phi$$) of quasiconformal deformations of $$\Gamma$$ with a homeomorphism $$\Phi:H^ 3/\Gamma \to H^ 3/G$$ homotopic to $$\phi$$ such that $$\Phi$$ ($$\lambda$$) represents an ending lamination of G (where we regard that $$\Omega$$ /$$\Gamma$$ is embedded in $$H^ 3/\Gamma$$ by pushing it into the interior). Especially if we consider a union of simple closed curves as a measured lamination, this result above implies a sufficient condition so that there exists an algebraic limit G of quasiconformal deformations such that these curves represent parabolic elements in G. This latter results is related to the result of B. Maskit [Ann. Math., II. Ser. 117, 659-668 (1983; Zbl 0527.30038)].
Reviewer: K.Ohshika

MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57M50 General geometric structures on low-dimensional manifolds 57S30 Discontinuous groups of transformations
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