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Cannon-Thurston maps and bounded geometry. (English) Zbl 1204.57014
Biswas, Indranil (ed.) et al., Teichmüller theory and moduli problem. Proceedings of the workshop at Harish-Chandra Research Institute, Allahabad, India, January 5–15, 2006. Mysore: Ramanujan Mathematical Society (ISBN 978-93-80416-00-7/hbk). Ramanujan Mathematical Society Lecture Notes Series 10, 489-511 (2010).
The central topic of the paper under review are considerations about results of J. Cannon and W. P. Thurston from an unpublished Princeton preprint from 1985 entitled “Group Invariant Peano Curves”. A result is the following. Let $$M$$ be a closed hyperbolic 3-manifold, fibering over the circle with fibre $$F$$. Let $$i: \mathbb{H}^2\to\mathbb{H}^3$$ denote the inclusion of $$\widetilde F$$ into $$\widetilde M$$ (where $$\widetilde F$$ and $$\widetilde M$$ are identified with $$\mathbb{H}^2$$ and $$\mathbb{H}^3$$, respectively). Then $$i$$ extends continuously to a map $$\widehat i:\mathbb{D}^2\to \mathbb{D}^3$$, where $$\mathbb{D}^2$$ and $$\mathbb{D}^3$$ denote the compactifications of $$\mathbb{H}^2$$ and $$\mathbb{H}^3$$, respectively. This was extended to Kleinian surface groups of bounded geometry without parabolics by Minsky in 1994. Then Bowditch in 2002 proved the above Cannon-Thurston result for bounded geometry surface groups with parabolics. The author gives a new proofs of all these results. They are extracted from earlier work of the author.
For the entire collection see [Zbl 1186.00038].

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 57M10 Covering spaces and low-dimensional topology 20F69 Asymptotic properties of groups 55R05 Fiber spaces in algebraic topology
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