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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.

Poincar\'e series

horocycle

limit set

Fuchsian group

rarefiedness

Discrete group

reduced function

minimal thinness

Kleinian group