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Cannon-Thurston maps for trees of hyperbolic metric spaces. (English) Zbl 0906.20023
If $$X$$ is a hyperbolic space in the sense of Gromov, then $$\partial X$$ denotes the Gromov boundary of $$X$$ and $$\widehat X=X\cup\partial X$$ the Gromov compactification. If $$Y$$ and $$X$$ are two hyperbolic metric spaces and if $$i\colon Y\to X$$ is a proper embedding, then a Cannon-Thurston map $$\widehat i\colon\widehat Y\to\widehat X$$ is a continuous extension of $$i$$. It is easy to see that if such a continuous extension exists, then it is unique, and the problem is therefore the existence of such a map.
In this paper, the authors define the notion of a tree of hyperbolic spaces satisfying the quasi-isometrically embedded condition. This is given by a path metric space $$X$$, a simplicial tree $$T$$ and an onto map $$P\colon X\to T$$ such that the inverse images of the edges and vertices of $$T$$ by the map $$P$$ satisfy several technical conditions. In particular, for each vertex $$v$$ of $$T$$, the subset $$X_v=P^{-1}(v)$$ of $$X$$ is a path connected rectifiable metric space which, equipped with the induced path metric, is a $$\delta$$-hyperbolic metric space, with $$\delta$$ independent of the choice of the vertex $$v$$. Furthermore, the inclusions $$X_v\to X$$ are uniformly proper. The notion of a tree of hyperbolic spaces satisfying the quasi-isometrically embedded condition is closely related to a notion introduced by M. Bestvina and M. Feighn [in J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)].
The main result of this paper is the following Theorem. Let $$(X,T)$$ be a tree of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition, let $$v$$ be a vertex of $$T$$ and let $$(X_v,d_v)$$ denote the hyperbolic metric space corresponding to $$v$$. Then, if $$X$$ is hyperbolic, there exists a Cannon-Thurston map for $$(X_v,X)$$.
The authors give several corollaries of this theorem. For instance, they give a new proof of Thurston’s Ending Lamination Conjecture for geometrically tame manifolds with freely indecomposable fundamental group and with a uniform lower bound on the injectivity radius. (The conjecture has been first proved by Y. Minsky.)
The authors prove also the following (also proved by E. Klarreich, in his Ph. D. thesis, SUNY Stony Brook, 1997): Theorem. Let $$\Gamma$$ be a freely indecomposable Kleinian group such that the injectivity radius of $$H^3/\Gamma$$ is uniformly bounded below by some $$\epsilon>0$$. Then, there exists a continuous map from $$\partial\Gamma$$ to the limit set of $$\Gamma$$ in $$S^2=\partial H^3$$. – Other applications concern graphs of hyperbolic groups, and the problem of local connectivity of limit sets of Kleinian groups. Finally, the authors describe examples where the existence of a Cannon-Thurston map is not known.

##### MSC:
 20F65 Geometric group theory 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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