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Weak concentration points for Möbius groups. (English) Zbl 0805.20041
A limit point $$p \in \partial D^ m$$ of a discrete group of Möbius transformations acting on $$D^ m$$ is called a weak concentration point if there exists a connected open set in $$\partial D^ m$$ whose set of translates contains a local basis for the topology at $$p$$. Every conical limit point is a weak concentration point. When $$m = 2$$, a complete characterization is obtained. In all dimensions, a sufficient condition for $$p$$ to be a weak concentration point is given. It shows that every parabolic fixed point is a weak concentration point, and consequently all limit points of geometrically finite groups are weak concentration points. Moreover, it shows that whenever the limit set is all of $$\partial D^ m$$, every limit point is a weak concentration point, and that for any group all but countably many points of the limit set are weak concentration 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) 57M99 General low-dimensional topology