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Limits of limit sets. I. (English) Zbl 1287.30007
The authors study the convergence of a sequence of Cannon-Thurston maps (CT-maps) defined on the limit set of a fixed Kleinian group. Let \(\Gamma\) be a geometrically finite Kleinian group. Let \((\rho_n)\) be a sequence of weakly type-preserving isomorphisms from \(\Gamma\) to geometrically finite Kleinian groups \(G_n\). The authors show that if \((G_n)\) converges strongly to a geometrically finite Kleinian group \(G_{\infty}=\rho_{\infty}(\Gamma)\), then the sequence \((\hat{i}_{n})\) of CT-maps from the limit set \(\Lambda_{\Gamma}\) to \(\Lambda_{G_n}\) converges uniformly to the CT-map \(\hat{i}_{\infty}:\Lambda_{\Gamma}\to \Lambda_{G_{\infty}}\). It is also shown that if \((G_n)\) converges algebraically to \(G_{\infty}\) and the geometrical limit of \((G_n)\) is geometrically finite, then the sequence of CT-maps converges pointwise to \(\hat{i}_{\infty}:\Lambda_{\Gamma}\to \Lambda_{G_{\infty}}\). The key to prove uniform convergence of \((\hat{i}_n)\) is to construct a certain function \(f_1:\mathbb{N}\to \mathbb{N}\) such that \(f_1(N)\to \infty\) as \(N \to \infty\) and such that, for any \(d_{\Gamma}\)-geodesic segment lying outside the ball of center 1 (the unit element) and radius \(N\) in the Cayley graph of \(\Gamma\), the corresponding \(\mathbb{H}^3\)-geodesic determined by \(\rho_n\) lies outside the ball of center \(O\) (base point) and radius \(f_1(N)\) in \(\mathbb{H}^3\) for all \(n\). For the case of pointwise convergence, a similar function which depends on each limit point is constructed.
As an application the authors give an alternative proof of the \(\lambda\)-lemma for the set of attracting fixed points of loxodromic elements in \(\Gamma\).

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