an:05236733
Zbl 1138.57020
Bowditch, Brian H.
The Cannon-Thurston map for punctured-surface groups
EN
Math. Z. 255, No. 1, 35-76 (2007).
00188442
2007
j
57M50 20F67 57N16 20F69 57M60 53C45
compact surface; hyperbolic space; Cannon-Thurston map; ending laminations
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.
Andrzej Szczepa??ski (Gda??sk)