×

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
PDFBibTeX XMLCite
Full Text: arXiv EuDML