an:06573170
Zbl 1375.20045
Jeon, Woojin; Kapovich, Ilya; Leininger, Christopher; Ohshika, Ken'ichi
Conical limit points and the Cannon-Thurston map
EN
Conform. Geom. Dyn. 20, 58-80 (2016).
00354849
2016
j
20F67 20F65 30F40 37C85 37F40 37F30 57M60
convergence groups; Cannon-Thurston map; conical limit points; Kleinian groups
Summary: Let \( G\) be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space \( Z\) so that there exists a continuous \( G\)-equivariant map \( i:\partial G\to Z\), which we call a \textit{Cannon-Thurston map}. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in \( Z\) in terms of their pre-images under the Cannon-Thurston map \( i\). As an application we prove, under the extra assumption that the action of \( G\) on \( Z\) has no accidental parabolics, that if the map \( i\) is not injective, then there exists a non-conical limit point \( z\in Z\) with \( | i^{-1}(z)|=1\). This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if \( G\) is a non-elementary torsion-free word-hyperbolic group, then there exists \( x\in \partial G\) such that \( x\) is not a ``controlled concentration point'' for the action of \( G\) on \( \partial G\).