### Abstract

A limit point p of a discrete group of Möbius transformations acting on S^{n} is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^{n} at p. For the case of Fuchsian groups (n = 1), every concentration point is a conical limit point, but even for finitely generated groups not every conical limit point is a concentration point. A slightly weaker concentration condition is given which is satisfied if and only if p is a conical limit point, for finitely generated Fuchsian groups. In the infinitely generated case, it implies that p is a conical limit point, but not all conical limit points satisfy it. Examples are given that clarify the relations between various concentration conditions.

Original language | English |
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Pages (from-to) | 285-303 |

Number of pages | 19 |

Journal | Topology and its Applications |

Volume | 105 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 |

### Keywords

- Concentration
- Concentration point
- Conical limit point
- Controlled
- Fuchsian group
- Geodesic lamination
- Geodesic separation point
- Kleinian group
- Lamination
- Limit point
- Möbius group
- Point of approximation
- Schottky group
- Weak

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Topology and its Applications*,

*105*(3), 285-303. https://doi.org/10.1016/s0166-8641(99)90064-0