Concentration points for Fuchsian groups.

*(English)*Zbl 0959.20046The local dynamics at a limit point of a Fuchsian or Kleinian group (a discrete group of Möbius transformations acting on the Poincaré disk or its boundary, the sphere \(S^n\) “at infinity”) can be complicated, and various types of limit points have been studied in the literature, among them the conical limit points. The authors call a limit point \(p\in S^n\) a concentration point if, for any sufficiently small connected open neighbourhood \(U\) of \(p\), the set of translates of \(U\) under the group action contains a local basis for the topology of \(S^n\) at \(p\). Various variants of such a definition are presented and studied in the paper, comparing them also with the notion of conical limit point (i.e. there is a sequence of translates of the origin of the Poincaré disk that converge to \(p\) and lie within a bounded hyperbolic distance of a geodesic ray ending at \(p\)). It is shown that for the case of Fuchsian groups (the case \(n=1\)) every concentration point is a conical limit point but that the converse does not hold, in general. Examples are given which clarify the relations between various concentration conditions, exhibiting also a different behaviour for finitely generated and infinitely generated Fuchsian groups with respect to limit points.

Reviewer: Bruno Zimmermann (Trieste)

##### MSC:

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57M50 | General geometric structures on low-dimensional manifolds |