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The Cannon-Thurston map for punctured-surface groups. (English) Zbl 1138.57020
Let $$\Sigma$$ be a compact surface with boundary components, $$(C^m)_{m\in{\mathcal P}}$$ indexed by a finite set $${\mathcal P}$$ which the author assumes to be non-empty. Moreover the Euler characteristic of $$\Sigma$$ is negative. Let $$\Gamma= \pi_1(\Sigma)$$. The subgroups of $$\Gamma$$ which correspond to the boundary curves are called peripheral. Let $$\Gamma$$ act discretely, faithful and type-preserving with no accidental parabolics on $$\mathbb{H}^3$$. Let $$N= \mathbb{H}^3/\Gamma$$.
The author proves the following: Let $$\text{inj}(N)$$ be half the length of the shortest closed geodesic in $$N$$. If $$\text{inj}(N)> 0$$, then there is a continuous $$\Gamma$$-equivariant map $$\omega: \partial\mathbb{H}^2\to \partial\mathbb{H}^3$$. Then the author proves that the map $$\omega$$ is the quotient of the circle by the equivalence relation arising from the ending laminations. The methods of proof are very technical. The author gives a wide point of view i.e. references to other works and many suggestions for further research which makes the paper especially interesting.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20F67 Hyperbolic groups and nonpositively curved groups 57N16 Geometric structures on manifolds of high or arbitrary dimension 20F69 Asymptotic properties of groups 57M60 Group actions on manifolds and cell complexes in low dimensions 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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