Discrete groups and thin sets

Journal article, 1998

Let $\Gamma$ be a discrete group of M\"obius transformations acting on and preserving the unit ball in $\Rdim$ (i.e.\ Fuchsian groups in the planar case). We will put a hyperbolic ball around each orbit point of the origin and refer to their union as the {\em archipelago of $\Gamma$}. The main topic of this paper is the question: ``How big is the archipelago of $\Gamma$?'' We will study different ways to answer various meanings of that question using concepts from potential theory such as {\em minimal thinness} and {\em rarefiedness} in order to give connections between the theory of discrete groups and small sets in potential theory. One of the answers that will be given says that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the set on the unit sphere where the archipelago of $\Gamma$ is not minimally thin. Another answer tells us that the limit set of a geometrically finite Fuchsian group $\Gamma$ is the set on the boundary where the archipelago of $\Gamma$ is not rarefied.