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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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