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Concentration points for Fuchsian groups. (English) Zbl 0959.20046
The 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.

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
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