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Geometric behaviour of Kleinian groups on boundaries for deformation spaces. (English) Zbl 0764.30036
Let \(\Gamma\) be a geometrically tame Kleinian group which is not isomorphic to a surface group, and which has no accidental parabolic elements. Let \(\{(\Gamma_ i,\varphi_ i)\}\) be a sequence of geometrically tame Kleinian groups in \(AH(\Gamma)\) (the set of faithful representations with discrete images from \(\Gamma\) to \(PSL_ 2\mathbb{C}\)) which converges to \((G,\psi)\) in \(AH(\Gamma)\) as representations. Suppose that for \(i\) the homotopy equivalence \(\varphi_ i\) is homotopic to a homeomorphism \(\varphi_ i:{\mathbb{H}^ 3\over\Gamma} \to {\mathbb{H}^ 2\over\Gamma_ i}\). Moreover, suppose that there is no parabolic element in \(\Gamma_ i\) on \(G\) whose preimage in \(\Gamma\) by \(\psi_ i\) or \(\psi\) is not parabolic. In this conditions the author establishes that the sequence of \(\{(\Gamma_ i,\varphi_ i)\}\) has a subsequence which converges strongly to \((G,\psi)\). This result appeared implicitly in W. P. Thurston [The geometry and topology of 3-manifolds, Lect. Notes, Princeton Univ.] though the author proves it in a different way form, using the notion of geometric limit in the sense of Gromov. Also he proves that the assumption that \(\psi^{-1}\) preserves the parabolicity is essential in the statement of the result above, generalizing a result of St. P. Kerckoff and W. P. Thurston [Inv. Math. 100, No. 1, 25-47 (1990; Zbl 0698.32014)].

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
Keywords:
Kleinian group
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