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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
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