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].

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].

Reviewer: Ken’ichi Ohshika (Osaka)