### 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 |
---|---|

Pages (from-to) | 285-303 |

Number of pages | 19 |

Journal | Topology and its Applications |

Volume | 105 |

Issue number | 3 |

Publication status | Published - 2000 Dec 1 |

### Fingerprint

### 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

### Cite this

*Topology and its Applications*,

*105*(3), 285-303.

**Concentration points for Fuchsian groups.** / Hong, Sungbok; McCullough, Dairyl.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 105, no. 3, pp. 285-303.

}

TY - JOUR

T1 - Concentration points for Fuchsian groups

AU - Hong, Sungbok

AU - McCullough, Dairyl

PY - 2000/12/1

Y1 - 2000/12/1

N2 - 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.

AB - 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.

KW - Concentration

KW - Concentration point

KW - Conical limit point

KW - Controlled

KW - Fuchsian group

KW - Geodesic lamination

KW - Geodesic separation point

KW - Kleinian group

KW - Lamination

KW - Limit point

KW - Möbius group

KW - Point of approximation

KW - Schottky group

KW - Weak

UR - http://www.scopus.com/inward/record.url?scp=16344377892&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=16344377892&partnerID=8YFLogxK

M3 - Article

VL - 105

SP - 285

EP - 303

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 3

ER -