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Cannon-Thurston maps, i-bounded geometry and a theorem of McMullen. (English) Zbl 1237.57018
Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2009–2010. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 28, 63-107 (2010).
Let $$G$$ be a torsion-free finitely generated Kleinian group, and $$G_0$$ a geometrically finite group which has an isomorphism to $$G$$ preserving the parabolicity and inducing a homeomorphism between the corresponding hyperbolic 3-manifolds. A Cannon-Thurston map is an equivariant continuous map from the limit set of $$G_0$$ to that of $$G$$. Cannon and Thurston showed the existence of such a map when $$G$$ is a doubly-degenerate Kleinian surface group corresponding to a fibre of the mapping torus with a pseudo-Anosov monodromy. It was conjectured by Thurston that Cannon-Thurston maps exist for general finitely generated Kleinian groups. Based on Minsky’s work, Klarreich showed the existence of Cannon-Thurston maps in the case when $$G$$ has bounded geometry (i.e. when there is a positive lower bound for the injectivity radii for the corresponding hyperbolic 3-manifold), and McMullen showed the same for the case when $$G_0$$ is a once-punctured surface group without assumption of bounded geometry.
Recently the author announced a proof of Thurston’s conjecture above for finitely generated Kleinian groups in general. In this expository paper under review, he explains his argument under the assumption that $$G$$ has “i-bounded geometry”. Fixing a positive constant $$\epsilon$$ less than the three-dimensional Margulis constant, a Kleinian group $$G$$ is said to have i-bounded geometry if there is an upper bound for the geodesic length of meridians on the boundary of every $$\epsilon$$-Margulis tube in the corresponding hyperbolic 3-manifold $$\mathbb H^3/G$$. This is a weaker condition than having bounded geometry. The author’s proof relies on Minsky’s bi-Lipschitz model manifolds which were introduced to prove the ending lamination conjecture.
For the entire collection see [Zbl 1213.35007].

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F30 Differentials on Riemann surfaces 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57M60 Group actions on manifolds and cell complexes in low dimensions
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