Concentration points for Fuchsian groups

A limit point p of a discrete group of Möbius transformations acting on Sⁿ 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 Sⁿ 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.