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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)
Kleinian group
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