## Abstract

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: ∂G → Z, which we call a 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 ∈ 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 ∈ ∂G such that x is not a "controlled concentration point" for the action of G on ∂G.

Original language | English (US) |
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Pages (from-to) | 58-80 |

Number of pages | 23 |

Journal | Conformal Geometry and Dynamics |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - 2016 |

## Keywords

- Cannon-Thurston map
- Conical limit points
- Convergence groups
- Kleinian groups

## ASJC Scopus subject areas

- Geometry and Topology