an:06750548
Zbl 1370.57008
Mj, Mahan
Cannon-Thurston maps for Kleinian groups
EN
Forum Math. Pi 5, Paper No. e1, 49 p. (2017).
00368308
2017
j
57M50 30F40
Kleinian group; Cannon-Thurston map; degenerate group
Question 14 of Thurston's famous and most influential problem list on hyperbolic 3-manifolds and Kleinian groups can be rephrased as follows. Suppose that \(\Gamma\) is a geometrically finite Kleinian group and \(G\) an arbitrary Kleinian group abstractly isomorphic to \(\Gamma\), by an isomorphism preserving parabolics. Then does there exist a continuous map from the set of limit points of \(\Gamma\) to the set of limit points of \(G\) taking a fixed point of an element of \(\Gamma\) to a fixed point of the corresponing element of \(G\)? Such a continuous map is called a \textit{Cannon-Thurston map}, cf. the paper of \textit{J. W. Cannon} and \textit{W. P. Thurston} [Geom. Topol. 11, 1315--1355 (2007; Zbl 1136.57009)], based on a preprint from 1985.
In a previous paper [Ann. Math. (2) 179, No. 1, 1--80 (2014; Zbl 1301.57013)], the present author showed that Cannon-Thurston maps exist for simply or doubly degenerate surface Kleinian groups without accidental parabolics, and in a second paper [Geom. Funct. Anal. 24, No. 1, 297--321 (2014; Zbl 1297.57040)] the author showed that point pre-images of the Cannon-Thurston map for such groups correspond to endpoints of leaves of ending laminations whenever a point has more than one pre-image.
``The aim of this paper is to apply the techniques developed in these two papers to extend these results to arbitrary finitely generated Kleinian groups without parabolics''. ``We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.''
Bruno Zimmermann (Trieste)
Zbl 1136.57009; Zbl 1301.57013; Zbl 1297.57040