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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

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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